What school in LA (Los Angeles?) are you going? Are you familiar with the areas in LA?
First things first. There are lots of room for rent on Craigslist with no lease (month-to-month). Unless you are super-rich and exactly know where you are staying, I suggest you go this route after a few days in the hotel. Once you know where you really want to stay, then rent an apartment, because apartment usually comes with 1 year lease minimum.
No. You can get the car first without insurance. Are you looking for brand new or used? Dealer or owner? There is Uber/Lyft to take you around when you are carless. Take advantage of airport shuttles.
Regarding Driver License:
I guess your FL license needs to be valid to be able to transfer, if not, you apply just like a new driver. I assume you have US passport and social?