Car is with the auto transmission specialist now. They are going to fit the transmission, including new rear main and torque converter seals, and new gaskets, filter and oil. They are also going to do a registration inspection so I can renew the registration finally.
It was interesting to drive it on the road for the first time since I drove it home after buying it. In drive the idle was pretty crap. It also felt quite slow, as if it was in too high a gear. I guess there is most likely more wrong with the current transmission than just reverse gear.
Looking forward to driving it home tomorrow with a new transmission!