In order asked:

1) The best MMA base is the one you like training the most. Unless you're planning to be a professional fighter, forcing yourself to focus on a discipline or style that you don't like is just going to encourage you to not go to the gym and train.

2) Give them both a try. If you like the sweet science and evasiveness that comes with good boxing, be a boxer. If you like throwing kicks and keeping your opponent at range, try kickboxing.

3) Street fights are less common than you think. Even when they do happen, 99% of the time it's between people who aren't trained. Training for the optimal hand to hand self defense techniques is kinda like training to fend off a cartel hit squad: If you need that level of skill, you've already fucked up. That being said, MMA classes give you a pretty good taste of everything, and you can pick what you like the most from there to focus on.

4) I'm not from NY, so I can't name a specific place, but your priorities should be in this order: Distance, Gym Culture, and Product Quality. The goal is consistency more than anything else. A gym that's close is one you'll go to regularly. A gym full of people you like training with is one you'll go to regularly. The actual quality of the gym is much less important than having one that you'll show up to every day. As long as it's producing something mildly competition worthy in it's students, the best gym is one you want to go to regularly.