You have a lot of different issues in the one message:
As a parent you often have to encourage your child to do things they may say they don't want to do - eat their vegetables, wash their hair, tidy their room, see their dad...
Hard as it is, You must encourage your child in seeing her father just as you would with any of the other things she might refuse to do but which you as a parent have a responsibility to ensure happens as part of your proper care for your child.
I would say your daughter sounds terribly confused by the situation. More time with her father may be the solution so she can see he is still a real part of her life and that she is important to him. At the moment she doesn't know what is going on. It is very hard to see straight when this is all new to you too i appreciate, but you have to try and step back from your own feelings and look at what is best for your daughter.
If he wants overnights suggest a gradual increase. He should really see her weekly, so suggest every saturday and insist he have that afternoon/evening time. This is not what is about convienient for him, he must make changes in his life too to do what is right for his daughter. If you can't come to an agreement with you then use mediation. Have you explained to him about your daughters distress, not wishing to sleep alone, night terrors etc. He needs this information to be able to think abou what is best for her in making these decisions.
It might be that your daughter is worried you are going to leave her too. Having more time with her dad will help her see that he hasn't left her life and maybe she'll be able to handle things more.
Maintence must be paid regardless of the contact arrangements, they are not meant to be related (though the current system only discourages contact by linking contact time to amount paid and is in serious need of an overhaul imo)
If you are going on benefits the maintenace issue will be irrelevant more than likely. If you are on IS you will only be allowed to keep £20 os whatever he pays.
Also your financial concerns re the mortgage etc need sorting out. You say you can't afford to pay it, yet you don't want to move and don't think you could afford anywhere else. It is very hard to deal with things like this when you are in the middle of it all, but you must get some advice and seriously look at your options. Go and see your CAB to see what they can offer you in terms of advice.
Do not begrudge your daughter the time and money her dad is now spending on her. He clearly feels guilty and doesn't know what else to do and wants her to have fun with him. If you can't afford to do these things try and be pleased that she gets to do them at all. Would you rather she sat around doing nothing but watching tv and being miserable with him?
Dad's parenting time is very limited so of course he will be doing the fun things and leaving the rest to you. Increase this parenting time and make it clear you want him to be an actively involved and equal parent and not a fun time only parent. And then go about ensuring this happens. Don't moan he doesn't do homework or telling off etc and then limit the time he spends being a parent so he will never be able to be an equal parent. At the moment he is pretty much relegated to being the equivalent of an aunt or grandparent who sees the child only occassionally and spoils them rotten when they do. If you want him to be a real parent then let him be. Sort out that extra time. Your daughter really needs it and from the sounds of it you need some time and space to sort yourself out too.
Increase the time to every weekend, add an afternoon/evening, two if you can. Agree that if everything goes well and he shows he can stick to this arrangement that in 3 months time overnights will start. Show willing and you will fare much better in court should he take it to that stage.
Take care of yourself, things will get better with time, but it will be a lot of time. List out all the issues and then put a plan together on how you are going to tackle each one (house, money, parenting plan etc). It might help you focus more clearly.
Stay strong
Gilly