Set up a time for regular chats, and have her initiate the call at that time. If she asks at random times, say "You can call Daddy at 5:30 like always and that way you know he will be home and available." I don't think you need to be more involved than helping her push the right buttons.
She's not walking around with the computer so there's not much he can see in your house/life. You say he will tear apart everything he sees - do you mean he critiques the decor to his child? That's weak. If he's that shallow and uses father-daughter time to discuss what she has hanging on the wall in her room or what color the paint is, she's going to stop calling him pretty quickly and then your problem will be solved. Don't get involved in it, don't respond.
If you're doing half the driving, that's already a lot - you've got 99% of the child care because he rarely sees her, so why are you driving too? And he's moving further away? My friend's ex was "too stressed" to continue his teaching job, so he arranged a year-long "medical leave" at full salary, moved 2 states away, and he wants her to drive. She said no and it's written into the visitation agreement that she got by filing for a modification of the prior order. (The dad used to take the kids to his house Wednesday afternoon through Friday - taking them to daycare or the school bus) and then he had alternate weekends as well. Then he moved to a remote area in Maine from suburban Massachusetts, met a woman, got engaged, and rarely sees his kids. His choice.
I wouldn't put up with the Thursday notification of whether he's going to see her. He's controlling your social life, and he's only seeing her when he's got no other options. Forget that. He takes his daughter and picks her up at X o'clock on his assigned weekends. If he wants to switch now and then, he can ask you nicely. But he takes her. If he doesn't show and she's sitting there with her little suitcase, that's terrible for her. If you don't want to face him/deal with him, then you do drop off at the police station where all the other contentious or abusive parents get their kids. There's never any drama there in front of the cops, believe me. If he persists in not showing up, then you petition for a modification and he loses time.
If he's having more than one emergency every 3 months (assuming he's not a surgeon or a firefighter), he's not into parenting. If he moved away and wants to forfeit the Wednesdays, that makes sense. He can come take her every other weekend, and perhaps he can take her 11-6 on one of the additional Sundays to make up for the Wednesdays. Or, since that cuts into your 2 full weekends a month, you can switch to a system where you get 1st and 3rd weekends, he gets 2nd and 4th, and if there's a 5th weekend, you alternate or he gets a long day on Sunday (e.g. 11-6). But stop this business of him deciding whether he's coming and you just sitting there waiting. If you don't want to deal with him, have your lawyer contact his lawyer saying that the schedule isn't being followed, so either he straightens up or you'll petition the court for a significant reduction, but you're not sitting around waiting to see if he has a better deal for weekend entertainment. Let the lawyers handle it - it will cost you less than the gas money you're spending to drive her everywhere.