It may be time to put things in writing all the time. Use email and some sort of notification program to verify receipt.
If he's entitled to the Tuesday nights per the order, then I'd let him have them. If that is detrimental to the child now that she is 7 (school work or other activities) then it needs to be revisited. If it mostly works for you, the only concern I'd have about taking it back to mediation is that maybe she'd encourage him to do something that's more for their benefit than the child's.
I am a stepmom and I would not tell my DH he can't talk to his child when it's not "his" time or tell him he can't contact his ex for things about the kids - that's called coparenting. The only times we told the sks they couldn't talk to BM was when it was inappropriate timing, like past bedtime. They could certainly call her later and talk, but not at 10PM. It wasn't about being jealous, but it being a bad time. Has your daughter tried to talk to him about how she feels when he won't talk to her?
The other thing is he is going to have to man up in his own home That's just something that he needs to deal with. I'm assuming reasonable communication between father and child/you and him here. Being the new spouse can be hard (I was much more wound up the first couple of years), but just based on this description it sounds like they're very insecure in the new household.
FYI, I rarely speak to my DH's ex. There's really nothing she needs to arrange with me. It's all between her/him/the kids. I give my input to DH. But DH and his ex certainly communicate about the needs of their children (mostly just SD now as SS is an adult). I would focus on a way to get him to communicate to you about your DD and for DD to resume reasonable phone contact with her father. At the end of the day, he needs to figure out his own compromise.