It sounds like you have it set to high and it is hitting the limit and knocking it back down. I have read about this happing when it gets cold out, but not the A/C. That said I can's see why the same wouldn't apply. I have the A/C Idle % set to 12 on my 96, but I'm not sure how that would work on a 1.6.
As for the battery I have no idea. I would put it on a trickle charger over night and see if it goes away. I do this to mine every once in a while as I read that given the nature of the gel cell that the alt. won't ever fully charge the battery if it goes low. This could be why it stays on for the first 10 min or so.
As for the ABS light does it turn off when the bat light does?