What happens to extra successes when treating a critical wound? Let's say I generate a net of 4 successes. Is the critical injury removed along with 3 regular wounds? Or is only the critical wound removed?

Extra successes on checks that take a long time generally mean that the skill check is performed in a shorter amount of time.

What happens to extra successes when treating a critical wound? Let's say I generate a net of 4 successes. Is the critical injury removed along with 3 regular wounds? Or is only the critical wound removed?

Not as RAW, I believe, because healing wounds is a separate roll. However, a Triumph or several advantages might let you remove a second crit automatically, or allow an attempt for a second crit.