Find first, what each cable is feeding, then it may make sense. Post back

I don't think it helps. The breaker feeds only overhead lights and
120V outlets. Specifically, I count:

5 overhead lights
1 outdoor flood light
14 120V outlets

Two of those outlets are on the ceiling in a shop area and power 120V
fluorescent lights. Two others power the two garage door openers.

Also, you should have 4 white wires you haven't mentioned. You have
4-12/2 circuits leaving the panel. Right?

4 12/2 cables tied to the two double-pole breakers. Each cable's
neutral goes to the neutral bar, and their grounds go to the ground
(which are separate since this is a subpanel).

Here's a part I didn't mention yet, but the more I look the more this
seems like the answer. What is leading me to investigate these is I
see that at some point previously the central A/C installers tied the
a/c power onto these breakers by sticking their wires under the
terminals with the existing ones (big no-no). Of course these devices
are 240V so they need a 2-pole breaker. I already know that these a/c
installers were utterly incompetent crooks, so I'm now thinking that
they went to the trouble of replacing the single pole breakers with 2-
pole so that they could feed the 240V, but of course didn't find
breaker space for what they needed and instead double-tapped the
breaker. This might account for the 30A rating on that breaker as
well, although other A/C units are tied to 20A breakers, so who
knows. I could see these guys just putting in a breaker until it
didn't trip it and saying that was good enough.

The reason I'm investigating all of this is so that I can fix it all
-- move the a/c's to their own 2-pole breakers, consolidate other
circuits as needed.

I wanted to ask about whether there was a realistic need for a 2-pole
breakers on individual 120V circuits and given your answers, I think
my guess above is probably what led to this fiasco. Now I need to
find out what the proper amperage rating is for the a/c units and fix
all of this.