How to repair a hole in wood laminate kitchen countertop

It may be possible to repair, but depending on the underlying material, I would take different approaches.

If it's particleboard, your best bet is trying to patch it with veneer. You can find sheets (for bigger patches) online, or banding at most big box stores. The main problems here are matching the current laminate and applying the patch so it isn't obvious. Some tips: Make sure the surface is clean and ready to take the patch, and try to make the patch as "un-square" as possible, so it doesn't stand out. Tracing paper and a lot of patience are required.

If it's plywood, I would refinish the cabinet. If the laminate problem is from heat, it's going to occur again after the patch. You can apply the same veneer patch method to plywood (it's easier), but this will likely be a recurring problem and you might look into a different cabinet finish or a higher quality laminate.

Sure you can. I'd recommend using a sharp Forsner type drill bit or sharp hole saw to avoid chipping the laminate. Cover the laminate with painter's tape and mark your hole centers. If your using a single 4" faucet fixture, use a template to drill the holes. If you are using separate handles/neck. the common spacing is 8 inches, but again, use the template.