What are the best practices for passphrase security?

Before getting to your specific questions, I want to say that there's a particular mindset that a lot of people have with security topics that you'll run into when trying to find information about things like this that I think it isn't very helpful overall: people love to fixate on doing security "perfectly", even though the weaknesses from a simpler approach might not actually be realistic.

As a simple example, it's common advice that you should never write down any of your passwords. From an objective standpoint, that's correct—not having passwords written down is strictly better. However, what's the actual danger? Is it a realistic possibility that someone's going to break into your house and look through your notebook to get your passwords? There's probably a very, very small group of people that actually need to worry about that situation. For almost everyone else, forgetting a password (especially if it's a complex one) is a far more likely "threat", and writing them down would have been better.

Of course, when it's reasonable, it's still always best to do things in the most secure way possible. But as an individual, unless you're doing something that's going to make you a specific target for attackers trying to get into your stuff, you really don't need to be too afraid of doing some things "wrong" if it makes it more convenient for you.

That being said, for most people that aren't extremely high-profile, the main realistic threat is that you register for some random site, its user data gets breached, and attackers take all the login info and try to log into other sites using the same username (or email) and password combinations. So the single most important thing is to never share login info between multiple sites. Unless you're being targeted specifically for some reason, this is probably the only thing that will ever cause you an issue.

So now, for your actual questions:

1. Do multiple encrypted drives need individual passphrases?

This is pretty much what I'm talking about - technically, it's probably best to use different ones. In reality, I highly doubt that you need to worry about some kind of situation where the attacker was able to get one of the passphrases but not the others. Using just one should be completely fine, the main important part is that your data is encrypted at all.

2 & 3. Is Diceware a good way to generate passwords, and better than my own words?

Yes, Diceware is fine. There's nothing about it that's magic though, if you want to use your own words that's going to be basically just as good. Again, the most important part is that you don't share passwords for multiple sites/purposes, and it's probably simplest to have some kind of generator that can handle creating a lot of random passwords for you. It's not particularly important whether it's Diceware, some other random-word-picker, or even just a standard password generator that uses random characters instead of words.

4. Do I need to add digits/mixed case/etc.?

No, not really. The important part is that you're picking passwords from a "space" that has a massive number of possibilities, to make it infeasible for attackers to brute-force your password by trying all possibilities. Making people use mixed case, numbers, and special characters is a good way to force them to expand their space, but it's not necessary. A long password that's all lowercase can easily be much more secure than a shorter one with a larger character set.

5. Can I use a single "root" passphrase?

No, you shouldn't do this. This is basically a slightly-obfuscated version of sharing the same password between multiple sites, which, again, is the single most important thing not to do. It's safer than sharing the exact same password, but it can be very easy for someone to see through, and I'd be surprised if some tools for analyzing login data don't already try to notice it. I can't remember the specifics offhand, but there was a significant security incident sometime in the past year that was blamed on someone using this type of method and the attacker figuring out their password based on data from other source(s).

6. If you use diceware-like passphrases, how do you deal with requirements to have numbers/etc.?

Overall, if all of your passwords are truly different like they should be, you're going to need some way of keeping track of them. A password manager is probably best, but writing them down is fine too (and make sure you keep a backup either way). So if you're keeping track of the passwords, it's not a big deal to just add some uppercase/symbols/etc. at the start or end to get past the restrictions. That won't weaken the passwords.