From my experience with a friend, I know that he cannot take her to a house with no fence around the pool. He also needs a crib or a mattress on the floor. He needs gates on the stairs and outlet covers and so on. He will need to provide proof of these things (photographs and/or receipts - photos are better because he could buy something and just return it). I don't think you have much choice re court at this point unless you can get him to agree to mediation which involves a specialist (sometimes a lawyer) but is cheaper than having 2 lawyers fight it out.
You need a support agreement and can have the state garnish his wages and send you the money.
Document everything - every payment he made, every time you found him asleep with her, every visit he had and every one he canceled.
My guess is the judge will say she is too young to be shuttled back and forth on a shared custody basis, although your ex might get joint legal custody (not physical) and the right to visitation. I can't believe that day care during "his" days would be cheaper than child support, so I think he is just trying to yank your chain and hit you where it hurts.
My friend's ex argued for visitation, but he didn't provide a crib (etc.) for the youngest, so the youngest didn't go. Then he wanted to make up the lost time, and she agreed because it was going to involve every Saturday night - as soon as the dad realized his social life was cut out, he backed down.
You need to get some sort of legal involvement, either thru legal aid or a mediator, because without an objective intermediary, all the conversations (and threats) are from him to you directly. My guess is that he is stringing you along.
Since you have no legal agreement in place now - from the sound of things - you are under no legal obligation to let him take her to his house. An hour car ride, 2nd hand smoke, a pool and no childproofing? I don't think you have too much argument about the dog - there's no correlation between the size of the dog and the danger, it's more the temperament and past behavior of the dog. However, if the dog has free reign and that means that the stairs aren't gated, that's your argument.
He can take her to the children's library, a local park or zoo maybe (near you), as long as he has a car seat for her, but otherwise I would limit the visits. I don't see why he has to sit in your house - he needs to be out of your life and also forced to create a life for his daughter. Meet him at the door with her in her jacket and with one diaper - and then he's on his own. You either shut the door or you leave and lock it. He wants to have her half time? Then he can manage for a half hour without you handling everything! Make it realistic (and therefore hard) for him, and be strong!