We will be staying in Carlisle and I don't know whether it would be cheaper/easier to rent a car for the day to get to the wall...and still be able to motor around a bit after to see some scenery, or to take the train to Hexham, take a cab to the wall and return the same way. Is parking a problem at the Wall ? We typically take public transportation everywhere we can...we will probably get an 8 day flexpass for Britrail. And York...are train routes sufficient to visit various sites of interest around the York area, and again, is parking a hassle?
To answer your parking question, and maybe some of the others, it looks like there is no problem. IMO a car would be ideal for the area.
It took me years and years of trips to Great Britain before I finally got to see Hadrian's Wall and the reason was the difficulty in getting there by public transportation so I highly recommend you rent a car. Parking is no problem at all at any of the forts. Just a long walk uphill. We visited Chesters & Housesteads forts, both a bit closer to the East coast but it's a very short drive all the way across. Why not pick up your car in Hexham and drop it in York? York is just about my favorite English city and worth a few days stay. I adore it and the York Minster, the train museum and even the cheesy Richard III. museum over one of the gates. Walking the ramparts is a real treat and you must visit the ruined fortress in the middle. I do oneway drops all the time in Britain and Europe. It's usually no problem at all.