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I change all my car and motorcycle tires myself, using a slide hammer type bead breaker, a couple irons and some good tire lube. The key is to get the beads of the tire squeezed together so they drop down in the center section of the rim, giving enough slack to get the bead off in one place without difficulty. Same with the new tire going on... make sure the bead is down in the drop center everywhere except where you're working. With stiff tires and narrow rims, I've occasionally had to get someone to stand on the tire to squeeze it into the drop center for me while I'm using the irons, but I can usually stand on one side while prying at the other and do it alone. If you try to pry a tire on or off without paying attention to the bead position, it can be very difficult, and even damage the tire bead.
To balance motorcycle wheels, I support the axle in a vice and make sure the wheel spins easily, then stick weights on until the wheel doesn't prefer to stop in one place when spun slightly. In my experience, this will balance them accurately.
I currently have Bridgestone BT45s on my pregen and they handle great, stick great, and give decent life. The miles you got from the rear tire sound good to me.
Putting new tires on probably fixed the wobble. When front tires get worn they often get a little squirrely. The is no "alignment" to do when installing tires, just making sure the bead is completely seated all around. Wire spoked wheels can get out of true and need truing to fix wobbles, but with our cast wheels, they don't go out of true unless they get whacked good.
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