If I put 2 explore counters on a location in the staging area with 1 quest point, is the extra counter put on the quest or is it lost?

I am pretty sure that the second token would just be overkill and not apply to the quest card. The only time there is carry over is when you actually put excess tokens on a location that you are currently TRAVELING. Not the staging area. This is my gut feeling; though, I may be wrong.

Yeah, as soon as the required Progress Tokens are placed on the location it is considered explored and would be discarded from the Staging Area. Any extra Progress Tokens that would have been applied are just disregarded.

You are correct - the only time tokens "overflow" to the quest card is when an effect puts tokens on the quest card. This includes the normal resolution of questing.

If something says to place tokens on a location, those tokens can only ever go on a location.