June 4th is Sunday, so it'll be sometime between 5 - 8 PM, right?

Yes.

If "up to 15 drivers on track," we should all be able to be on the track at the same time?

I hope so, which is why I would like to make reservations.

Is "5 minutes of qualifying" not the same as 5 minutes of practice?

Yes. For $75 you get 15 minutes of track practice, then 5 minutes of qualifying to determine your position for the race, then you get to race for 15 minutes.
15+5+15=35 minutes.

Either Super Race or Super Grand Prix, does everyone get to spend all 20 min. or 35 min. on the track?

Yes

I have not done this go-kart racing before, but I'm assuming that even 20 min. of track time still goes by really fast?

Yes, it will.