Do Daken's claws tear through his skin like Wolverine's claws do? If so, shouldn't the claws bonded with the Muramasa metal have caused the wounds in his hands from popping the claws to not heal? Was this brought up at all?
The writers ignored that particular detail regarding the claws up until the very end. The last time he and Wolverine fight, when they get ripped out, he's left with two giant scars on his arms that don't heal for quite some time. But you're right. It still doesn't explain why he didn't have to bandage his wrists after every fight he used them in.
Word of God is that the claws were encased in a sheath that protected his skin from the metal until the claws were fully extended.
That's not just Word of God, it was explained in the comic when he first added the Muramasa metal. To quote the Tinkerer: "Since the metal cannot come into contact with your soft tissue without catastrophic effect, I've manufactured these sheaths out of adamantium. They will be anchored inside your forearms."