[Q] Will my Fave Apps Still Run After I've Flashed a New ROM? topic

This must be so insanely obvious that I'm being a total idiot by asking. However, I have searched both this site and the rest of the Internet 30 ways to Sunday and haven't even come close to an answer. I have several favorite apps that I use. But my Nexus 7 (first gen) is running like a freaking turtle, so I'm thinking about rooting it and flashing a new ROM so that it is once again functional, at least for reading and the occasional exercise video. But if Overdrive, Nook, Taptu, Feedly, Kindle, Google Books, Pocket and Food Planner Pro don't generally run on non-stock Android, then it would really be a waste of my time. I expect there may be a certain amount of "It depends" in the answer, but is there a way to find out without cluttering up the forums?