Yep, you can do that, but we actually found it to be more economical to keep it the whole time rather than dealing with transportation coming and going. You will have to try all the companies to find the best deal. Good luck! (We got a minivan for 8 days for about 260.00 total)
Avis allows you to rent at MCO then drop off near the Port - they shuttle you to the port, only a few minutes away. We did this last year (Carnival) and it worked great. Remember to give yourself plenty of time, there were a lot of people doing the same thing. It didn't take a long time since they had several shuttles running, but I wouldn't wait until late in the day (we went around 11:30 or 12, didn't have to be at the ship until 3:30, I'd say we were on the ship before 1).