Ok, all things considered, here's my impression:
You certainly have a suction side air leak. It probably is a water leak as well, but is not leaking very much (for a while it sounded like you weren't sure you were losing any more than evap). One of your drains is clogged; it seems unlikely that you'd have a broken pipe just there, since the drains are probably teed together halfway between them. When you blew air and eventually got bubbles out of it, I think you didn't get the clog out, but managed to loosen it a bit so that the air could get by.
Most suction-side leaks are either in the pump basket lid (probably not your case since switching to skimmer gets rid of it, but check anyway) or in the above-ground pipe before the pump (have you thoroughly checked all those connections?). Assuming your equipment is above the water level, having a leak in the above-ground pipe would suck lots of air but not lose much, if any, water. A leak below the water level (underground, such as a cracked connection or broken pipe) would suck air, though usually not that much unless its surrounded by coarse material like gravel backfill. It will leak water.
What puzzles me about this is that you suck LOTS of air, enough to lose prime, but you're losing very little water. That sounds like it almost has to be an above-ground connection leak.
As for the other pipe in your skimmer, try blowing some air in it with your compressor. That'll tell you if it goes anywhere. have a pipe in the bottom of my skimmer that doesn't do anything.