OK, here's my thought. He isn't showing that he cares about you or your marriage. I'm sure he has good qualities, too, and that there are good things about him/ your marriage that you want to hang on to.
I do understand feeling like you have no idea what you're supposed to do. I worked with my husband for a while and I felt like that all the time. I'm betting he probably felt like you do.
I think you should enumerate all the tasks you do in a day, from brushing kid teeth to washing dinner dishes to things as little as clipping tiny fingernails. Then write how often these tasks must be done. Assign them a day. Then put your name by some of them and his name by some of them. Make a list of things you need in your life, like time to take a bath and write a list of things you husband can do with the kids while you do your stuff. Don't just write "Go outside," Write "Play soccer outside" or "weed the garden with the kids."
Sit him down and explain to him the gravity of the situation and make it clear that you and/ or your relationship cannot last like this. Pull out the list and explain it all very clearly to him. If he hemms and haws, bust out a suitcase, pack up YOUR things only and check into a hotel at night. You might want to have another list of things he needs to do for the kids that night prepared, just in case. He'll keep your kids alive. He'll be scared out of his wits, too, and my guess is he'll be more responsive the next day. My sister in law had to do this with my brother and it worked. It was extreme, but so was her unhappiness.
I am not trying to sound condescending by telling you to write things out as if he were a 5 year old. I'm telling you because I needed someone to do this for me while working in a well drilling company where I only felt like men knew what was going on and I was in the way. I ended up contributing a lot to the organization but I had a lot of learning to do first. And I needed encouragement and positive feedback without criticism.
When he does a crappy job at cutting nails, don't criticise him. Tell him "thank you for cutting the kids' nails" and leave it at that. If he feels like he did something right, he might attempt to do it again. Maybe practice will make him better! The point is you want to build up his self-confidence so he'll not feel defeated and unappreciated (I'm sure you're in touch with that feeling!).
Key to all of this might be making it clear that you cannot be in a relationship that is all give and no get. But do not make any threats that you will not stand behind. Maybe a reward system would work. Think of him as your child, I guess. He clearly wasn't raised to be responsible around the house. Now you've got to raise him right.
If all else fails, try getting on that show "Wife Swap" and hope that a no-nonsense feminest gets him for a week and sets him straight. (I'm kidding of course.)
Good luck.