I’m assuming you already have your airline tickets, but if not, you would be better to fly into Porto and out of Lisbon. It would save a day on the train. You haven’t left much time in Porto by training there from Lisbon. But, other than that you itinerary looks pretty good for your limited time.
You definitely don’t need a car for any of the places you mention. Sintra is easy on the train from Lisbon. And, both Porto and Lisbon have very good public transport.
Regarding places to stay. I like close to the Rebeira area in Porto and Chiado for Lisbon. There are many hotels and apartments in both areas and they are close to lots of restaurants, shops, etc.