Beheeyem

Say you have Silver Mirror attached to one of your non-EX Pokemon, and it is the only remaining Pokemon. If your opponent's Beheeyem tries to attack with "Damakinesis", the six damage counters obviously don't get moved to the Pokemon with Silver Mirror since Beheeyem is a Team Plasma Pokemon; but do the damage counters get removed and then discarded, or do they not move at all?

The non-EX Pokemon with Silver Mirror attached is a valid target for the choice of where to move the damage counters to, but since Silver Mirror prevents all effects of attacks done by Team Plasma Pokemon the damage counters would be discarded rather than attached to the Defending Pokemon.

Can I choose to move less than six damage counters for Beheeyem's "Damakinesis" attack?

If you have six damage counters in play, you must move six of them.

Do the six damage counters for Beheeyem's "Damakinesis" attack have to all come from the same Pokemon, or can I move a total of six from multiple Pokemon I have in play?

You can choose as many Pokemon as you have in play, moving a total of six damage counters. If you have less than six damage counters on all your Pokemon in play, you move all of them.