If the condition of a card effect is the "first E" does that mean it must be the first E of the combat phase or just the first E that you play.

For example: My opponent was playing Dan as his character whose second effect reads:
First E: Commit all characters on the field.

If I were to play an attack then activate one of my E's immediately after that, would my opponent then be able to activate Dan’s effect since it was the first E that he played this turn, or would he not be able to use it since I activated the first E of the phase?

Also, does first E mean it could be the first E of each attack or must it be the very first E played during combat phase?

First E means it has to be the first enhance you activate during any attack. It doesn't matter if it's the first of the combat phase or the first attack of the phase. It just has to be the first thing you enhance with. Kind of like the First Form ability has to be the First Form you perform during that turn.

Stamp for Kilik850.

Your timing to play a First E: is at the first possible opportunity for you to play such an ability. If it's your attack, you get the first crack at enhancing and it can be a First E: but if you pass, you lose the window to play a First E: anymore on that attack. If it's your opponent's attack, they get to play an enhance first, and then you get your shot at a First E: (whether or not they played an Enhance themselves). If you pass, you lose the chance to play a first E: to that attack.