Try a "thin" coat of E6000 glue.
Cork does not need to be glued. If it is compressed in the bottle neck, and then made wet, the water makes the cork expand.Have you tried super glue or hot glue gun? Not sure what you are trying to...
You could use a hot glue gun or water based contact cement
Cat toys, kindling, starting BBQs.
No
Titebond would be fine.