I hope the car company are a bit more specific about their location rather than "in the parking lot". Is there going to be a representative meeting you at the exit after customs?
Typically I would advise against staying in Palma if you have a car as it can be a hindrance however as you have such a short time to visit it does make sense to stay in Palma. With a car you will be able to visit Valldemossa and Soller at your own leisure (visit Valldemossa early as the nearby small car parks fill up rapidly). If you're not looking to linger you could visit Valldemossa for a couple of hours, drive to Soller and have lunch and a leisurely stroll (It's not a big place) before driving on to Port de Pollenca and maybe even fitting in the old town of Alcudia (not the port). You could even have dinner there or return to Palma for more choice.
The following day I'd recommend setting out early and visiting the town of Arta and the spectacular caves. You'll see a lot of advertisements for the Caves of Drach but my advice is to avoid those as they are a disappointing tourist trap, the Caves of Arta are much better. Dependendant on time I would the follow the coastal route south to Cala D'Or. This is a town that was designed and built in the Ibizan style, lots of low level, whitewashed buildings. It's not an old town and it is a popular tourist destination however at this time of year it'll be a bit quieter. There are lots of good restaurants here or else you could visit the nearby small fishing port of Portopetro, pleasant but quiet at this time of year or move on to the larger town of Santanyi. If you don't like driving through narrow, old town roads then you may wish to park on the outskirts and walk into the centre however be mindful of where you parked as it's easy to get lost! Santanyi is popular with German tourists so expect to be addressed a lot in German but English is widely spoken (but not always) and of course Spanish (or Catalan to be precise). If you're lucky there might be a market on that day in which case you'll be spoilt for choice with an incredible array of food and drink.
The direct route back to Palma will take you on the MA19 which is a very easy drive and once past the roadworks near Campos you'll be on the fast road direct to Palma. It'll be a whirlwind itinerary but will make the best use of your limited time and will give you a taste of the mountainous north, the coast and the warmer, drier and flatter south. I hope you enjoy it as much as we do.