Hi Jin,
We just got back from Florida (Ft. Lauderdale with our 8 mo. old son. We did not get a seat for our son because we knew he wouldn't sit in it anyway. I pre-reserved seats in the aisle and window. On the way there we had someone in the middle who was more that happy to swap the aisle for her middle, but on the way back we had the row to ourselves. I'm still breastfeeding, so I just fed him during take off and he went right to sleep for the majority of the flight! Since your daughter is 22mo. I would suggest lots of snacks to keep her chewing so her ears don't bother her - it can be VERY painful(gum for you since you aren't used to flying either). Bring lots of activities that won't take up a lot of space like coloring and reading. It should only be about a 2.5 hour flight.
We thought it was great that the rental car company had carseats so we wouldn't have to take ours.......DON'T DO THAT!!!! The seat was not the right size, it was dirty, we had no clue how to install it and the staff was not very helpful & I think it was missing parts!!! If you are going to check your seat, the suggestions the other moms had were excellent - heavy duty plastic bag, carseat cover, and/or a box. Just make sure you know the dimension restictions your particular airline has on checked luggage (they will charge you if a box is too big!!) You can also check the seat at the end of the jetway, right before you board, where you check your stroller. These "gate checked" items that also include wheelchairs (so you know they are extra careful with them) do not go in the same compartment as the luggage or cargo and it will be at the gate when you arrive in Florida instead of being thrown around on the baggage belt. It's very important to make sure you have identification on the carseat, bag or box just in case! If your carseat or stroller becomes lost or damaged, the airline does provide "loaner" items until yours are recovered or replaced.
If you do have a seat for her on the plane and are taking the carseat onboard, make sure it is FAA approved, otherwise they will not let you use it in flight. Hope this helps. Sorry it's so long! I used to work for an airline at PHL International Airport so I know what goes on behind the scenes!!
~J.