Say that I have a start or end of turn effect that requires me to play a card (such as La Capitan's La Paradoja Magnifica, where at the start of the villain turn I play the top card of the villain deck). In the event that the card I play also has a start or end of turn effect, do I activate that too?
According to the FAQ:
If a card comes in to play at the end of a villain turn, and it has an "end of the villain turn" action, does that action happen this turn?
Yes. You are still in the "End of Turn" phase, and all "End of Turn" actions in play must resolve for that phase to end, even if those cards came in to play as part of that phase.
It seems safe to generalize this to the start of the villain turn as well.