It's really not about his skill level at fixing stuff - it's about his unwillingness to take responsibility for basic things that come up in the course of daily living. It doesn't take skill to call someone to pick up an old mattress, right? I'd be furious about this also. It's good you are recognizing it now before it goes on for six months and you are stuck in a pattern where you do everything.

Does he take responsibility for any aspect of your shared life? Bills, cleaning, food prep, cars, pets, etc? Is there anything that's his task?

Every time I reread the comment about you calling up the trash people because you're female, I cringe. I'd be very tempted to tell him to go back to his old place since he still has it!