I was going to say "check steering neck bearings, rear shocks and shock bushings" along with "check tire pressure and balance" but it sounds like your problems are deeper than just checking for bearing or bushing wear.
+1 to the "take it to an independent pro and have them evaluate it and recommend fixes." Rob at Meteor Motorbikes in Emeryville might be able to help.
I'll be frank with you; that bike is worth about $4000 on its best day (more/less depending on mileage/farkles.) You're likely to spend several thousand chasing these issues out of it. It may make more sense to find another of the same year/make/model and just port your accessories over to it. Keep that path in mind when looking at the estimate and choose wisely.
Good luck with the repair. I'd be going back to the shop that did the rear tire with this story and asking what they'll do to make it right, personally.