I cannot figure out how to get claude to give me his quest, talking to him dont do it, so uh what does do it?
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Provided you haven't already completed it, you need to progress the main story by completing Venesh/Sarrina's quest by speaking to the butler in your home base, either rescuing Rose with Venesh or freeing the person that is imprisoned in the Crusader base with Sarrina. Once you've done that and finished exploring around the areas you unlock from doing so, going back to your home base will cause a message from the butler to pop up to speak to Claude which moves the storyline forward.
If you've reached the point after beating the demon that there's the king's messenger waiting back at your home base in the main room and the warning of "proceeding further will lock out other options", then you've finished Claude's quest.
It depends on which snake woman you're talking about, as there are several. The common thread between all of them is the ability to apply the Frozen effect instead of being able to just stun them and no immunity to the Burn and Harsh Chill DoT effects. The Ancient Serpent also has no immunity against base stat debuffs like Defense and Magic Defense lowering effects. I tested the Crossroads update with Esthera, NaN, and Kushiel in the party so it is possible to down them though using Jin and Pierce for tank and healer may make things a lot easier depending on your party level.