Given I am from out of town and not having my own tailgate, would it be best for me to just take a cab from my hotel to campus on gameday? Are cabs easily available in Baton Rouge?
A cab to campus is easy, one getting back after the game would be very difficult. I don't live in BR anymore, but there are shuttles. I don't know if one goes to your hotel or not. You should call your hotel and ask them what they would recommend.
Before game day is usually better. There won't be nearly as many people. Look through this main forum, there are several posts similar to yours from people making their first trip to Tiger Stadium. Search the forum for "first trip" or similar and you will find tons of info.
Am I safe to assume that on game days, there are no strict alcohol rules around campus? Open container...etc?