Opponent A has Frey Hospitality as their plot card

Opponent A attacks Opponent B with an Intrigue challenge.

Opponent C supports B and steps in with support but loses the challenge.

I know Opponent B must pay the claim ( intrigue challenge ) but which opponent pays the Frey claim effect, is it C ?

Daenarys said:

Since C lost the challenge and Frey Hospitality hits "that player" who loses the INT challenge, it's C who is on the hook for the plot's passive effect.

To add another similar example could you please clarify

Opponent A plays Frey Hospitality

Opponent B plays Twist of Fate

Opponent A does a power challenge and wins against Opponent B and changes from a power challenge to an intrigue. As well as the card claim is Frey also applied ?

Opponent A has Twist of Fate

Opponent B has Shireen Baratheon ( F26 ) as a character agenda

Opponent B attacks A with a military challenge and wins and subsequently changes to a power claim. Does her response also kick in ?

You're making a mistake in your interpretation of Twist of Fate; you do not change the challenge type after you win. you only change the claim effect. So, to respond to your questions, you cannot trigger responses to winning a challenge of a type that you didn't actually initiate just because you resolved the claim of that type.