When a headhunter asks if I’m currently in the process, do I tell them with which fund?

Generally I would say the less detail you give HH the better. It's not the most ethical busines and many of them will try to use any info they have in any way with no regard to how that impacts you. Look at your position - it is not completely aligned with theirs. They may use any information you give them to benefit themselves to you detriment. So as a rule they don't deserve details unless those details help you negotiate somehow.