There are many OL cards, game effects, etc. that allow an action or a trap or an event to be played if the heroes have declared a move action and are moving.

If a hero has an ability, a skill, etc. which allows him to move up to his speed, or provides additional movement points (beyond his actions), are the above cards, events, traps, etc. activated if he uses those movement points, or are they not allowed because he is not actually doing a "move action".

This has become a serious bone of contention within our gaming group. Any insight would be appreciated.

If the effect specifically triggers on a "Move Action" then no, other skills and abilities that provide movement points are not a "Move Action" and you can't do it.

However, if an effect triggers on the hero "entering a space" (or words to that effect) then it doesn't matter why or how the hero has entered the space, just the fact that he has done so is enough.