Will having a 1" gap really be that awful? It is a finished vanity vs a builder's built-in, so it will at least look clean at the ends.

Construction adhesive But I highly recommend that you brad nail every other row or two.

I'm going to guess that the continual running is due to a worn out flapper not providing a good seal. They will wear out due to the chlorine in the water supply and there are chemical resistant models. Try putting some food coloring in the tank and see if the water is coming from the under-rim ports.

Try this. Drill a hole with a bit that is several sizes smaller than the final size. This will be your pilot hole.

Using final size bit, slowly drill from "back" side (side that isn't seen as much) to a depth of half the thickness of the wall.

Then drill from the front side and finish the hole. Presumably, there will be less material being pushed out compared to fully drilling from only one side.