Can seeking asylum be used as a reason for bringing charges against someone? As far as I understand, exercising legal options cannot ever be interpreted as an illegal act. I think it's much more likely the UK will pick Assange up on behalf of the US, who have threatened to charge him.
Yes. Failure to surrender is a separate offence. The Met has already said that he's still wanted in connection with that charge, but they'll be scaling back resources as it's a much less serious offence than rape.
Assange's bid for asylum does not give him immunity, under British law, from normal criminal charges (which is how they view the Swedish charges). Witness their promises to arrest Assange if he left the Ecuadorian embassy, regardless of his subsequent right to leave for Ecuador after being tried.