The rule says "Each character may only receive one Medicine check each encounter...."

Does it mean a PC has to choose between a Medicine check to heal Wounds and a Medicine check to heal a Critical Injury in the current encounter or he can attempt both?

You can do both, because critical attempts point out it's one attempt per crit, so clearly more than one are possible. You want to do Woumds first an maybe get a Triumph an take care of a crit all at once.

The one check per encounter is in the paragraph discussing wound recovery.

So does that mean, if a character performs a medicine check on an ally during a combat scene he's not able, by RAW, to perform an after combat healing? That seems strange, and kind of contrary to most narrative uses of such stuff. There's the "plug the hole" method during the bullet storm moments, but then after, they are able to focus and start doing some serious medical treatment.

Is the "plug the hole" basically "use a stimpak" in this system? Seems kind of unfair to the character who decides to go doctor/medic/healer, to be so limited in the thing he's good at, when everyone else can do their special snowflake stuff all day.

The structured encounter rules point out that an action or skill check does not automatically equal one turn in a round and can take longer if a GM chooses.

Also wasn't really addressing any notion of fair, just answering a question.

I'll have to find it, but there is a reference to an encounter equating to 12 hours when in downtime.

A Session should always be made up of multiple encounters. Thus, if an encounter that involved combat has ended, a new check should be allowed. I know for me, an encounter is "over" when the situation has resolved. Sometimes that's one combat, sometimes that's multiple combats that are all connected. It all comes down to context.