“I mean who decides on what court to go to in the first place?”
I’m probably not the best person to respond to this because I remain a “minarchist” myself, but there are various anarcho-cap responses to all of what you said.
First, when you said “won’t the rich [and powerful] just have an advantage?” – they certainly do now, and nobody expects absolute utopia.
Secondly, police and the courts might be two separate businesses (then they won’t be in collusion, necessarily at least).
As for “who decides which to use?” the way I’ve always heard it described there would have to be mutual agreement between the parties (or their advocates), and that if this sounds “not plausible, or anyhow the guilty party just won’t agree to any and will string it out,” the people who have worked this out have proposed mechanisms to insure that won’t happen, and they point to historical examples of similar institutions that they say have worked in practice.
But it’s probably best left to a full Anarcho-cap to explain their position. I know Robert “Bob” Murphy has given Mises lectures on this, one is here and another is here (note: they cover pretty much the same ground, having been given at two separate times to two different audiences; I mention this because you might only want to watch one, depending on how much time you have).