With only one night at your disposal in Taormina you can’t do all those things.
I prefer to stay in Giardini as it is less expensive and more likely to have parking on premises. I then drive up to Taormina and park in the huge multistory garage called Porta Catania and walk to the city center.
Check hotels with parking using the booking.com filtering options. I don’t remember the name of the one where I stayed, it’s been a few years.
Driving from Taormina to Palermo is super easy. It’s all freeway very similar to interstates in the US. I prefer the coastal route via Messina, along the north coast. It’s a bit longer than the Catania-Palermo freeway but less traffic and very scenic (when you are not inside the many freeway tunnels). The driving time is about 3.5 hours from Taormina to Palermo airport, plus pit stops. You can break the trip half way with a short visit of Cefalù. it’s very easy drive, I wouldn’t do it by bus or train if you already have a car. Too much hassle to drive to Catania airport, drop the car, then take the bus to Palermo. Keep the car and drive directly.