For the first question, I'm fairly certain you can't use a resource if it's not listed on the character card. Kim can't use romance. I guess in theory you could use cards with the 'any resource' icon to BUY cards that give romance but thematically it wouldn't make sense. I don't think it would be much help to do so anyway.
For your second question: look at the thin rectangular bar under the challenge card's name. If it's labeled Music then it's a music challenge.
well, that surprises me. I am certain everyone can use everything. The two resources on the character card just seem to indicate some synergies with your starting deck and the combos you can perform. No reason she should not be able to use romance.