I need some help with the sciency bit of my short story

I'll start with the usual caveat that accurate science does not always make for the most exciting stories, so I agree you shouldn't worry too much about the actual distance. Choose a distance that works best for the story (how much warning do you want Earth to have, for dramatic purposes) and come up with a reason for detection at that distance.

To get a rough sense of the distances, technically speaking we probably have the ability to measure the gravitational effects of an object that size past the orbit of Neptune (see this article on a hypothetical Planet X for example). However, those gravitational effects 'accumulate' over the course of many orbits of such an object, so probably would not be relevant for an object only recently entering the solar system and heading right for Earth.

In that case, assuming the emissions from the ship, the best chance for detection is via observing it optically through its occlusion of the background stars. We have the capability to essentially image the entire night sky every couple of days, so we can assume that a detection could theoretically occur once the object is large enough to obscure stars in these images.

Using the information here for simulations of upcoming observations from LSST, we can expect it to be able to image around 11 billion stars over what is a little more than half the night sky. Assuming a uniform distribution, this works out to very roughly an angular separation of ~20 arcseconds between stars in these images. Thus, we would want your hypothetical ship to appear large than a few tens of arcseconds to be noticed. Let's say 50 arcseconds. The diameter of your craft is ~19,000 km (Earth's diameter is ~12745km, by the way. your number looks off). For an object that is 19,000km long to appear with an angular size of 50 arcseconds, it would have to be roughly 80,000,000km away. This is about 0.52 Au, or about half the radius of Earth's orbit. For reference, Mars's current distance from Earth is about 0.62 Au, so we probably wouldn't detect this object until it was closer than Mars.

This is assuming that it is noticed as soon as it covers more than a few stars in the data, but given how much data is produced by such a survey, it is likely that this object would not be noticed until it was much closer, but its approach could be traced by looking at archival data. This is all back of the envelope of course, but it provides a starting point for further refinement based on additional properties of the craft. If it emits strongly in any wavelength it could probably be detected from much further out.

‘Oumuamua

This object was much smaller than your planet ship but was notable for its speed and angle. Basing the story on the discovery of that object would probably be the most realistic.

Would your starship be entering the solar system from a direction that objects commonly arrive from? In that case we have more resources pointed in those directions. But if it is coming from a unique direction that might also give it earlier attention.

The search for Planet X would also give you useful data. If there is a dark Neptune-sized planet somewhere in the Oort cloud that can only be detected by gravitational inference then you might be able to extrapolate the distance at which a similar earth-sized planet ship might become visible to us.

There are a number of instruments studying the outer limits of the solar system although of course none of them are dedicated to this purpose. I believe the mission to catalog dangerous asteroids has ended and the new missions have been delayed or canceled.

But in terms of present-day capability, depending on the albedo of the approaching planet ship and whether it is being piloted in a way to increase its stealth, I’d wager that the Pluto orbit would be about when we would pick it up.

From a technical point of view (if you want to lean more towards hard SF), this will depend on filling in some variables.

Real variable: how much does the ship mass? Is it a hollow shell? Is it a Matryoshka setup (shells in shells in shells for max surface area kind of thing)? Does it have some exotic setup like being powered by a black hole which would give it a huge amount of mass disproportionate to the exterior dimensions?

If you want to put a number on the total mass, and are also willing to say 'this mass behaves like normal mass for gravitational calculations' then it becomes pretty straightforward to calculate when and how much the grav becomes detectable.

Detection from visual observation: pretty much only going to happen from noticing the occlusion effect (blocking starlight from behind it) since you mention it is completely black - unless it is radiating IR? Is it giving off any radiation anywhere on the spectrum? If so, that is detectable and would give you some wiggle room to play with for when someone on earth notices it by detection method X or Y.

You would also want to decide if your alien vessel is going to interact gravitationally with the local environment. Do you want to do those maths, and stick to them for consistency, or would you prefer to use a plot device of 'alien tech mass alteration / gravity control' to not have to deal with it?

Naked eye visibility: either pretty darn close (near-ish moon close) or it would need to get in between earth and the sun (visibility by occlusion).

Impact on earth? If you have them interact with gravity normally, then massive tidal force stuff, tidal waves and such, if they get close (moon-ish close, within an order of magnitude or less by my back-of-the-mental-napkin math). If you plot-device gravitational interaction away, then them getting in-between earth and the sun from close enough to block enough sunlight would start impacting us: cooling effects, reduced solar power, premature winter...

btw 25,512 is not 1.5x earth's diameter, 1.5x earth's diameter would be closer to 19,000km (I think you put down 2x earth's diameter)

To make a short answer out of a long one, it depends on how fast the ship is. If faster than light, so not based on real physics, then it can be detected once it stops moving at light speed, plus the time it takes light to travel from that point to earth.

If less than light speed, then once it is close enough for its gravity to have affected something we monitor, like an outer planet or planetoid. Gravity effects will propagate forward from the ship at the speed of light. So I would suggest in this scenario, that the first sign of it is from detected anomalies in the orbits of distant planetoids around the sun.

If the ship isn't entirely hollow, it will have substantial mass. If you're following even modestly hard sci-fi principles, any acceleration under its own power will be very noticable for two reasons: (1) Any non-fantasy drive will have massive emissions. A reaction engine the size befitting a planet-class starship will be bright, no matter the underlying principles. Plain rocket? Gigawatts of power emitting from the exhaust, visible from basically everywhere. Photonic? Gigawatts of power emitting from the exhaust, though with enough technology this could perhaps be narrowly focused onto retrograde, thus possibly hiding it from earth. Fusion torch? Gigawatts of power again, visible from any aspect. There's no way this exhaust isn't there for a reaction engine. The question is whether it can be hidden from earth, as with a very high tech, narrowly focused photon drive. (2) If you turn on the drive, you're producing massive amounts of waste heat. Hell, you also generate slightly less massive amounts of waste heat from any onboard systems like computers, sensors, life support, etc, as well as sunlight. But the drive is undoubtedly much more intense. That heat needs to go somewhere. Either your ship heats up and radiates it into space, which would be visible. Or the ship dumps it into a "thermal battery", which would be heavy and have limited capacity. In any case, even the most high-tech drive can't fire forever without emitting at least thermal IR. As an alternativel, the ship could have any of these designs of thermal radiators to intentionally dissipate heat. Those are, to my mind, all Hard SciFi approved, except for the exotic spike. Some of those spray/dust/plasma radiators could befuddle the astronomers observing them quite a bit - they're basically magnetically contained beams of hot particles that cool off outside the ship, then are looped back around to be caught, reheated and reemitted. You'll be able to detect the element of the particles by their emissions, so choose something sneaky. Perhaps a perfect black-body. Or abundant, like hydrogen ions. So the stealthiest hard sci-fi solution here would be a photon drive with a tight beam, with any waste heat being dumped as emissions of hot hydrogen or hot graphite. That way, there can be some ambiguity about the origin of the weird emissions.

In either case, the capacity of such a ship to adjust its trajectory will be severely limited if they want to stay undetected. If they're coming in from interstellar space, they're coming in fast. If they want to stop, they can do one of three things to adjust their trajectory: (A) Turn on the torch, telling everyone what's going on. (B) Impact a planet-sized body, smashing it to smithereens, also telling everyone what's going on. Hopefully surviving the impact. (C) Gravity assist / swing-by maneuvers to slow down. These are actually sneaky, but slower and predictable. Swing-bys of venus and/or Saturn and Jupiter would probably be post-hoc identifiable in the astro data /u/gpl mentioned, but certainly not outright detectable a priori around Saturn and Jupiter. Certainly out there, you'd only get a single obscured star every few days, which makes the track hard to see. If all you have is a track of a shadow obscuring stars, any swing-bys, once discovered in archival data, are a clear sign of intentional behavior. They also clearly show you the distance to the target (which is otherwise hard to ascertain) and its future trajectory. Oh, and swing-bys are basically planned long in advance, so once that thing is on its way, it'll continue along its path of swing-bys until completion. And once Earth has identified the trajectory, they can figure out where the sequence of swing-bys will take the ship. Ohh, and one more thing: Those swing-bys will disrupt the orbits of any lighter planets it passes by, which could also be not-so-sneaky.

Oh, and once earth starts looking in the right part of the sky, this thing will be unmistakable. Focus JWST on it, and there's no hiding anymore. The difficulty is, to borrow military terminology, target acquisition, while target tracking is easy.

This is, of course, all assuming you're not going for a physics-defying gigasphere kind of spaceship.

(Edit: I suppose this indirectly addresses question 3: There is no "stopping your movement to remain undetected". Stopping your movement, unless planned long in advance as part of a careful and long sequence of swing-bys, means to be give your position away.)

Pretty fuckin far. Two pieces of good news for you: (1) a whole bunch of the characteristics of the ship will go into determining this. This all already depends on some suspension of disbelief, so basically you could get away with almost whatever distance you want. (2) Astronomers like to say that if your number had approximately the right number of zeros on the end, you're close enough.

My suggestion would probably be to just not explicitly call out a distance.

I don't have direct answers answers but you need detail for some of your specifics:

Assuming that the object is on a very slow path towards Earth, at what proximity will its effects be known?
At which point will scientists observe its effects, view it, or detect it with instruments?
If that is at all possible, what would be a threshold in which the "Round Borgs" would have to interrupt their movement in order to remain undetected?

Does it have the density, or the mass of a spherical cow in a vacuum? It probably won't be like our rocky metallic planets, but if you can figure that out, it would inform some things about distance and physics.

At which distance will it be visible to the naked eye (if at all)?

Depends on its reflectivity, but even Mars, a big dull ball of rusty dirt, shines at night. You could have it be a small spot that some astute stargazer or interested astronomer notices with limited suspension of disbelief.

And if they chose to get dangerously close to Earth, what would be the impact on our environment?

Depends on mass again. It would basically be a living ball of stuff, so quite massive, so there would be tidal effects, weather effects, and an absolute catastrophe if it's too close, as in unpreventable planetary collision/attraction, and/or orbit destabilization. A consideration is that the barycenter, the center of mass, between the earth and moon is roughly just below the earth's crust, and the moon is roughtly 60% as dense, and 1/4 the size of earth. So it will be very observable, and and exert an immense force at surprisingly far distances. We would probably notice it on this side of Jupiter if it's not otherwise observable because it would black out the sky where it is unless it either cloaks or somehow manages to slip between any points of light we can track (cloaking would be the best cover to side-step this), and any gravity readings on our planet. Systems like LIGO. which are specifically for gravity waves, but are finely tuned for the current environment, would likely start to exhibit anomalous readings unless it somehow counteracted its own mass.

Also consider the Square Cube law which partially holds that any creature would have to be significantly stronger at a larger size, and so more dense, and heavy, to support itself. That ship would have to be massive for its size, more like an elephant then the cow I jokingly use. Especially at its size, as it will have to resist its own gravity, unless you somehow want to hand-wave it or solve it.

If you need more of this sort of stuff, here is a good forum for all sorts of topics.

world building stack exchange