I'd recommend taking it to a guitar tech and letting them take care of it. It will take you longer to figure out the problem, and then fix it, than it will for them to fix it completely.
I'm not familiar with that tremolo, but, if it's anything like a Fender, where there is a block that extends through the body, you can often take a wedge of wood and push it behind the tremolo block from the backside of the guitar, locking it into place. If your bridge is a floating tremolo, where you can pull up and push down, this will stabilize your tuning nicely. It's common practice.
An alternate way to go about it is to reduce the spring pressure in the tremolo so it no longer is balanced and is pulled against the back of the tremolo cavity by string tension. That's how I have the tremolo on my Ibanez strat. It stays in tune nicely, single note bends act like they should, and sustain is as good as the guitar can do.
As for replacing the bridge, yes, it's possible. They'd remove the bridge, cut some wood to fit the shape of the cavities that remain, glue them in place, do some sanding and filling of any gaps, then shoot matching paint over it all. On top of the work you'd have to buy a fixed bridge and tailpiece.