If I have more prizes remaining than my opponent and I play Black Belt, but then I KO one or more of my opponent's Pokemon before I attack, would I still be able to use the effect of Black Belt if I'm no longer behind on prizes?
Yes, you can. The requirement of being behind on prizes is just for playing the card, and once it's in play you can use its effect even if you are no longer behind.
When using Smeargle's "Portrait" Poke-POWER, can I choose a Supporter that will have no effect?
You cannot pick a supporter you couldn't normally use; for example you can't choose "Twins" unless you are behind on prizes at that time, nor Aaron's Collection if you don't have anything in your Discard Pile. You would have to choose a different Supporter if one is available.
When Yanmega tries to attack, do you evaluate the number of cards in your hand for the "Insight" Poke-BODY before or after discarding a card for Giratina Lv.X's "Invisible Tentacles" Poke-BODY?
The Insight Poke-BODY is evaluated before your attack even begins; therefore it is checked before discarding any cards for Invisible Tentacles.