After the opportunity attack, the creature resumes its action. [Emphasis mine.]
It doesn't say that the target gets to complete its action, only that it gets to resume it. However, now that the target is stunned (or, as you point out, dazed, if it has already taken another action), the action it was performing is now invalid, so upon resuming the action said action now simply fails. I think this is what they were trying to express with that last bit about "If the target is reduced to 0 hit points...", but they could have been more general to make it more clear.
The same would happen if an AoO were to render a target immobilized, for example -- it now can't move, so its movement action now fails.
No, standing from prone doesn't trigger booming blade's damage.
I don't see any specific rules definitions for "movement." Based on definitions in other editions, my opinion is that a character "moves" when they walk or use some other "special type of movement" such as jumping, flight, swimming, etc: when they move from their original position, rather than just standing up, dismounting, or otherwise reconfiguring their posture but remaining in the same place.
The spell doesn't say "if the target willingly takes any action." Therefore, attacks and environment interactions are probably fine. The spell doesn't force you to remain perfectly stationary like a statue. Given this, it seems clear to me that doing things that use movement but don't change your tactical position don't count.
Some places in PHB190-191 use the phrase "deduct... from your speed" instead of "spending [X] feet of movement." This might be a clearer way of looking at it. Some actions, such as standing up, reduce the amount you can move in the turn even though they are not "movement" for the purposes of triggering spell behavior.
For a useful corollary, it makes sense that mounted movement would still trigger the effect, even though you're not using your own movement or deducting from your speed.
Although it doesn't affect the lack of definition in 5e's rules as written, D&D 4e says "Whenever a creature, an object, or an effect leaves a square to enter another, it is moving." 4e rules don't apply to 5e, but they provide insight into the background of D&D rule design.
Best Answer
The Booming Blade effect triggers before the opportunity attack.
Booming Blade states:
Opportunity Attack states:
Emphasis mine
The creature first begins to move, triggering the Booming Blade and then, right before it leaves your reach, the opportunity attack triggers.
Of course this is all assuming you have not spent your reaction, your reach is 5ft and the enemy attempts to immediately leave your reach instead of moving around you.