I was playing my second game last night (BattleLore has become an instant favorite of mine), and my opponent was trying to counter after he was forced to retreat one hex, which I advanced onto. He stated that since we were still adjacent to each other, the counter is viable. The way I understand it, however, is once a retreat die is rolled, that resolves before a counter can take place. Then, since I advanced, I am no longer on my original hex (and neither is the opponent), so a counter cannot happen. Is that correct?

You are correct. Both the attacker and defender must remain in their original hexes in order to counter. If they retreated and you advanced, they cannot counter.

If they retreat and you Pursue and attack them, they can counter that second attack as long as they don't retreat again.

What Budgernaut said. If you need a rules reference, go to: Reference Guide, pag 3, under Counters. Bullet point 2 is the one you could need to show your friend