The loneliness of Jodie Foster

Why didn't they investigate the scene of the crime and see the big button on the wall?!

Why the Russian fiance plotine?

Humanization for Hank's character. It shows he had his own life, and talking about it with Peter gave viewers a look at their relationship outside of the main plot. Also gives a glimpse at how gullible and desperate he actually is to fall for a romance scam like that.

Why the domestic abuser incident flashback when it was ultimately pointless ?

If you mean the Wheeler case where Navarro shot the guy, disagree it was pointless. It's the origin for the rough dynamic between Liz and Navarro, because Liz knew Navarro's temper towards culprits could be lethal. Murder is a step beyond the typical rogue cop who goes too far. The fact she was willing to cover it up anyway, and not force Navarro into retirement, shows she's also not strictly a "by the book" officer herself. Them covering it up was also ultimately used to try to dissuade Liz from investigating the Tsalal case any further.
One other reason I realized while replying though: it shows the limitations of the police to intervene with certain situations due to the larger systems. They'd gone to that house multiple times and KNEW that there was domestic violence and abuse occurring, but they couldn't actually intervene because the victim refused to press charges. They had no legal grounds to arrest him, so they could only watch and wait. This limitation ties into the overall season's themes and ending.

Why did a group of PhD's instantly go berserk and murder someone without hesitation?

For the first guy, it was heat of the moment anger because she'd ruined a good deal of their research. Then she lashed out at the next guy to get near her (I think he was trying to help/check on her, but I'd have to watch the scene again), at which point the rest seemed to join in partially due to rage, partially "self defense" because she attacked them (in self defense). Basically just felt like it escalated super fast and they were moving on instinct/panic, not going berserk.

Why was Prior so calm about killing his dead, cleaning up his corpse, and disposing of him?

Shock. Dude's brain basically shut down as a defensive measure so he could do what had to be done, and then break down later.

Why waste time on the vet completely misdiagnosing the method of death?
Why did the scientists have weird creepy wounds and damage to their bodies that ultimately didn't matter?
Why include Rust's dad long aafter he was supposed to be dead and make him do a weird ghost dance?
Why are the ghosts and hallucinations essentially real in universe?
How did the tongue get there? It makes no sense.

Putting these all together because they're all related. This season (admittedly the only season I've seen so far) seems to be toeing the line of supernatural and non-supernatural. I'm thinking the implication is meant to be that ultimately most of the case is mundane, but that there was a bit of influence from spiritual forces to nudge events to unfold the way they did.
Alternatively, I'm wondering if there was some sort of natural gas or something churned up by the mining operation that caused hallucinations. The wounds were self-inflicted if I remember right, so seems likely they were hallucinating something horrifying before dying of hypothermia. Don't know if there's any natural gases that would fit the overall symptoms, but it'd be a more rational explanation for the mining disaster Otis was caught up in too.
It still wouldn't explain the tongue though, so I think that was meant to be supernatural and a sign of Annie's spirit seeking justice.

Why was Navarro's likely suicide basically celebrated?

That was an open ending related to the above supernatural themes. Apparently the implications are that either she died having found some sort of peace with herself; she just left Ennis and is now living off the grid (maybe at the lake house Liz visited at the very end); or that she's now straddling the line between the living and the spirit world thanks to her maybe-supernatural visions. (Last one came from an article with Kali Reis I looked up afterwards.)

Why is vigilante mass murder straight up celebrated?

Now, for my personal take and opinion: I don't think there would have been any justice otherwise.
The mining company's influence tainted the case from the beginning. They had strong influence over the police, as directly shown by that superior from Anchorage going along with the "slab avalanche" story and pressuring Danvers into dropping the case. Before that, they also covered up Annie's murder by having Hank, a police officer, move the body and hide a tip about her being in a relationship with Raymond Clark. No one knew about Hank, but most people suspected the mine was involved in her murder somehow. Tension had also been building for a long time between protesters, the mine and the police.
When the women found the evidence that the scientists had covered up the pollution, they had no reason to believe the police would help. Trying to raise a case through legitimate means would likely just have it covered up and buried, or else dragged on for years before anything was changed, just like what often happens in the real world.
The murders were the women snapping because they felt a need for justice after delivering multiple stillborn babies and now knowing for a fact that their home was in danger, but also knowing that the proof they had wouldn't really change anything.
I think Danvers going with the slab avalanche story after learning the truth was a little nod from her about how corrupt the overall system is and that she knew the case wouldn't have been solved either way. Even at the end, she secretly leaked the recorded confession about the pollution online rather than submit it officially with the implication that it would have been buried otherwise. It ties into what I said about the Wheeler case showing how police ultimately have limited power within the larger systems at play.

What was with the one-eyed polar bear?? Because when it first showed up, it had the air of a Character instead of just a random polar bear. I thought for sure it'd be tied to Danvers' son's death or the overall plot beyond her son having a one-eyed polar bear plushie.

He's in the background of the video being taken by the other guy when he suddenly gets some sense and says "She's awake" or something. The season tries to push the supernatural, and then sort of walks it back and seemingly wants to straddle the line of supernatural, to maybe appease those who simply don't believe in the supernatural or don't think it belongs in True Detective, while using supernatural elements to lazily push mystery and intrigue in ways that would be harder to write without it. I even saw it coming by episode 3 that the tainted water was going to be used to explain all the weird things happening, the hallucinations people are having and what not, and even though the finale did incorporate it, they barely even covered it and just broad stroked it and essentially doesn't actually try to explain or justify it in any way.

So with the finale explanation of the cleaning women orchestrating this whole thing, how is it that Raymond Clark happens to be the only one who manages to avoid the attack? Either the show wants us to believe he actually has some kind of supernatural sense, or he got stupid lucky that he had a hallucination or psychotic break that has nothing to do with the cleaning women just outside about to attack the scientists.

I also found the part where they find Raymond Clark at the facility in the end a bit strange, and once they secure him, the first thing they do is torture him by making him listen to the video of Annie dying. At this point, they don't even know that much about what happened or his involvement. They know he just attacked them since he trapped Liz in the freezer and hit Navarro with the extinguisher so in theory this could be used as an explanation, but it seems a bit over the top considering that they're chasing him with guns and after what happened to the rest of the scientists he would obviously be a bit on edge not to mention from their perspective he probably wasn't seen as mentally stable exactly seeing as they saw what happened in the above mentioned video. Jumping straight to torturing him with the sound of his ex-love interest before they even know what involvement he had in her death is just strange.

The other thing that bothered me was the pacing, like I know part of the show is detectives who take these cases personal and go above and beyond the job, maybe the other issues with season 4 bias me on this, but I didn't seem to have this kind of issue with pacing in the other seasons. It was just unusual how much they pushed the case in every circumstance, with literally no concern for personal lives at all. Like I get it, that's a personality flaw that can make people interesting, in season 1 Rust basically had no personal life and was working the case nonstop pretty much, but he wasn't a total lunatic attempting to ruin others lives by making them like him. Meanwhile, Liz literally does not care about Prior's personal life, and Prior seemingly doesn't care for this brief window of time and then has some weird explanation if you can even call it that about how he doesn't want to be the smiling idiot. Again, I can get that it can be an interesting personality flaw, but it just felt so pointless. Like there was no real clock ticking in most of the circumstances, so it was like purposefully destructive just to keep the pacing fast and ensure there was always some easy tension available.

The other part that I found strange was Hank was pretty straightforward to Kate (the mine company owner) that he wasn't a killer, and then all of a sudden he just shoots the engineer guy right in front of Liz in a brazen execution. Like at that point he's basically committing to not only killing the engineer, but likely killing Liz (or what else is he going to do?). Now one could certainly argue that he already helped cover up a murder by moving a body so it's not too far a step removed from killing someone himself, along with his personal life issues happening at the time, but it still felt like a weird amount of escalation for no reason. Again like the pacing thing above, it just doesn't seem to have a compelling reason to push the pace or escalation like this. If he's ready to convert into being a killer, and he already got away with following her in such an obvious way and apparently she didn't seem to notice (the finale mentioning he awareness of him following was only something she would have seemingly realized after it all happened), he could have just followed her out into the middle of nowhere and killed them then rather than deciding to do it in her house.

Also one more thing, it was a seemingly cheap tactic or twist how the ice cave was right under Tsalal station, and as they are pointing this out on the map or talking about ice caves, there's no mention of it ever being near Tsalal station or directly under it actually. Felt like it was intentionally omitted to be part of a cheap twist or story development without giving too much away too early.