2. Top Card effects happen until they are no longer top card. However, once I play my next card to the right, does it immediately become the Top Card or do I resolve the Top Card effect one last time as the next card enters play? This would make the most sense to me since the Obari set has top effects that happen on next card you play.
I think this is really a very technical way to lookt at it: I would say that the top effect is basically an ongoing effect not something that gets "triggered". So by that logic, when a new card comes into play there is a short window where it is still the top card but a new card is in the middle of "coming into play" and that is where the top card can actually use its effect.
3. Actions and reactions require that you flip the card afterward. However, what if you have an effect that does not allow you to flip that card? Can you still do the action and don't flip the card or does it prevent the action from occuring? Also if I have an effect that makes me flip a card and I can't flip that card, is that legal? Or does the protection against flipping only prevent opponent effects from flipping my cards?
I think i read that somewhere already. But common sense would tell me that "flipping" is the cost of something; if i can't flip something i.e. pay the cost, i can't use its effect.
The last one was answered in the FAQ on the back of the rulebook:
Q: If I am required to flip a card of my choice, can
I choose a card that cannot be flipped?
A: No, a card that cannot be flipped is not a legal
target for a flip effect.
4. Do cards that activate when you flip a card activate from using action/reaction cards?
Not exactly sure what you mean by that. But if a card effect is triggered when a card is flipped, i'm pretty sure it doesn't matter why the card was flipped.
5. How many cards are in the core set?
Also found in the rulebook under "Components: 1 Core Set (118 cards)