"Should I follow up with a thank-you note after an interview? I know you mentioned it can be helpful to discuss what you liked about the interview. Should I do this right after the interview?"
Yes, you should definitely, when you can, follow up with a thank-you note after the interview. You don't always need to write to the actual partners or people you spoke with, but to the recruiting person. You should say something along the lines of, "I enjoyed talking with these people and appreciate what they said about this," "I'd love to be part of that," and something along those lines. And if you say those sorts of things, that's very positive and that jump out at the firm and make them much more enthusiastic about hiring you.
You don't always need to write directly to the person that you spoke to. It's often very useful if they hear what you wrote from someone else, but people do that. People always want to hire people that want the job. I've written a lot of thank you notes, and I do believe that you don't always need to write to people directly. If the word gets that you wrote something nice to someone else, that can be much more helpful than doing the opposite.