Your plan is doable and you can live reasonably if you're in Alabama where the winters are not 'too' harsh. However, you'll probably have a big propane bill regardless. Also, since you'll probably have to pay your own electric you have a lot of months that you'll need it...and the animals will need it.
You said you've found a place to park so have you checked out if they'll accept a long-term renter? Will they accept your dogs? Are you sure they would not be barkers if left alone? This could be a problem, if so.
You would probably have to have someone deliver the RV to the site unless you have a truck that can tow it.
If you can pay cash for the RV that would be a big plus. Definitely look for a used one. If you and your girlfriend decide to go separate ways, will you be able to afford it on your own?
You are not the first young person to do something like this but get every future expense down on paper and make sure you can swing it. You'll need to register it and get insurance. Propane and electric will be major expenses. The rent shouldn't be more than an apartment, in fact, much less. Other daily expenses should not increase.
You both will probably be on somewhat different schedules for school and jobs so RV size shouldn't be a big issue. You should probably be looking at around a 30'. Good luck!