Never heard of a fiberglass gasket. It's probably just the inner fiberglass skin, or what's left of it. You're either going to have to remove the inner skin or the outer skin to get all the rotten wood out. I can almost guarantee you aren't going to be able to find a pre-fabricated fiberglass skin, and if you did, I wouldn't use it, as it would be incredibly difficult to get it to have as good a bond as you laying your own glass.
Lag bolts look like big wood screws with a bolt head, and are what hold the metal motor mounts to the boat hull. Usually they go into raw wood and are not sealed, so it's a potential entry point for water to start rotting the wooden mount pads, which if connected to the stringer/bulkhead network and the transom, can cause those to rot too.
You said the motor mount stringer "seemed pretty solid". How did you check? If you didn't drill a hole and look at the wood shavings, you're not really checking.