If I play a card as part of a start/end of a turn effect, do I apply their start/end of turn effects as well?
According to the FAQ:

If a card comes in to play at the end of a villain turn, and it has an &quot;end of the villain turn&quot; action, does that action happen this turn?
Yes. You are still in the &quot;End of Turn&quot; phase, and all &quot;End of Turn&quot; 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.