Should you choose your committee members or should your advisor?
Choosing your committee is a collaborative process ideally, and should take place with input from your advisor, but also accounting for your own preferences.
The first thing you should do is check about the constraints on your committee put in place by university rules. This can be quite complex, and will often limit the freedom of choice you have in the selection process - for example, your hypothetical "Degree is in CS, everyone but your advisor is from English" example is simply outright impossible at the institution I got my degree from.
Is it more important that your advisor already know your committee
members or that you do (e.g., your advisor knows them but you've never
met them before vs. you know the member but your advisor doesn't have
a strong or pre-existing relationship with them).
Someone should know your committee members. That can be you, or that can be your advisor, but someone should know if they're likely to be problematic for this particular dissertation (it's a theory dissertation and they hate theory, etc.)
Is it better to get someone in your discipline or someone doing more
related work (e.g., if you're getting a computer science degree, you
ask computer science faculty vs. you're getting a computer science
degree but everyone except for your advisor is from the English
The "perfect" committee member is someone in your discipline doing related work. When those people don't exist, you should probably aim for a mix - you want a committee that can go "Yes, this is clearly a project worthy of a degree in $Discipline", but also people who can provide input on the specifics of your project.
Your advisor can probably talk to you about people who "should" be on your committee for various reasons, including political ones ("It will look strange if Y isn't on your committee...")
Should you pick people who are already invested/interested in you
succeeding or is the dissertation process supposed to be a chance for
you to win people over to your side?
Your committee should be people already interested in your success - believe me, even people really excited by your work and interested in you moving forward can cause problems. Someone whose an outright skeptic, and might be inclined to just dismiss the whole project? That is not someone you want on your committee. You have the rest of your career to try to win people over to your side after you have your degree.