I'm a stepmother of 2 girls and a mom of one son. My husband and his ex did not have a good relationship at all, and coparenting was virtually nonexistent. The girls were with her most of the time, and we had visitation probably similar to what you have. The kids are all adults now so Skype wasn't an option. We had to install a phone for just the girls and their dad because his ex didn't want him tying up her phone to talk to his kids. So I think it's great that your SD Skypes with her dad.
As you've realized, being book smart and having good grades has absolutely nothing to do with emotional growth or maturity. Taking her for counseling was an excellent idea, but if it can't be sustained because there's no cooperation between households, I can see why the behavior is going on. But it's not happening with her friends or in the classroom, right? So it's TOTALLY within her control.
I think your SD is doing the baby talk and reactions because she's getting a payoff for it. She gets attention (you talk to her, her dad talks to her) and so on. Negative attention is still attention. Of course it's completely enabled by what you say her mother does with the baby talk. I guess I would prioritize - if she wants to carry around a doll, fine. She may need a security symbol. Transitions from Home A to Home B are tough for a lot of kids. My son (raised with both his parents in a good marriage) still took items from upstairs to downstairs, and vice versa, as some sort of connection to "unite" the sections of the house. So I'd let that go. So if she brings something from one house to the other, let it go. If she wants a new purchase, I'd have her earn the money and buy it herself. If she doesn't have an allowance, you might consider instituting one for doing age-appropriate things with your family, a few chores (making her bed, setting/clearing the table, helping with meal prep, and other tween/teen things). Then she can decide where she spends that money - if she wants to spend it on a baby doll, that's her choice. But then there's no money for other luxuries.
The bursting into tears? I'd ignore it. I'd grab a towel, mop up the spill in a nonchalant way, and move on with the conversation wherever you left off. ("So, as I was saying, I think the weather will be great tomorrow so who wants a picnic in the park?") If she continues to blubber, you and your husband can either both get up and go to the next room to discuss the plans to go to the park, or tell her she can go to her room to collect herself, and you'll look forward to having her back when she's gained some control. End of discussion. Don't keep going on and on - she's getting too much attention for it already. I think you may need to say, "House Rules: no tantrums, no fussing over small things." I would suggest you do NOT say, "You may get away with that in your mother's home, dearie, but not here," as incredibly tempting and true as that may be!
I do think it's okay to say that Movie X or Activity Y are for older kids who can control their emotions. "We can't have that kind of crying in the movie theater, dear, so we'll go another time when you can go all weekend without a meltdown." If she expresses interest in doing something, great. But let her know it's contingent on her behavior. Does she do this out in public at all? If so, I would march her butt into the car (out of the restaurant, out of the museum, whatever) and head home. Same as you would do with a 3 year old. No endless discussion, no analysis...just, "Sorry, that behavior isn't acceptable. We're leaving."
She's going to want privileges appropriate to her age and grade level very soon. You have the control where that is concerned - but ultimately, SHE has control over how much fun and how many privileges she has with you.