1. If I use The Only Game that Matters, affecting just a single Stark character, is it possible for the Stark player to cancel the effect using Fear Cuts Deeper Than Swords? I believe it isn't since I don't really choose a target. Still I would appreciate a second opinion.

2. I'm pretty sure I know the answer to this one, but just to make sure.

Moving power tokens for Power claim and discarding cards for Intrigue claim takes place after the 3 step of a framework action window, right?

1. No, Fear Cuts cannot be used to cancel a general effect that only happens to hit a single character. Fear Cuts requires a character to be CHOSEN as the only target. If the card doesn't say "choose" (and Only Game doesn't), the play restrictions on Fear Cuts cannot be met, no matter how many cards "feel the sting."

2. Claim effects resolve DURING Step 3 of resolving the "claim effects" framework event.