The minis only come with 1 climb and 1 dive card. But to my understanding a plane is allowed to do 2 climb maneuvers in one turn ie. a climb, straight, climb. So since there is only 1 climb card how can we do this?

That there is only one Climb card in the deck should suggest to you the answer specifically: No, you can't climb more than once in a turn.

And you can't do an Immelmann in the same turn as a climb, nor a Split-S in the same turn as a dive.

In short, you can only play one climb turn a turn, and/or one dive card a turn. Er .. but you can do an overdive wihich is a special move to lose 2 altitude (Stall -> Dive -> Straight (treat second straight as another dive))