They grow out of the gassy thing. Our pediatrician explained that they have to learn to relax so they use the right muscles when they poop. Eventually they figure it out. As for the sleeping, Ds was like that too at that age. We just kept trying and he started sleeping better as he go older. I wore him a lot so that I could get other stuff done. The first several weeks are just really hard.
Talk to your dr about gas drops or gripe water. That may help with some of the pressure. It really is dangerous to put them asleep on their belly until they are rolling back to front and front to back.
If you can watch her the entire time she's sleeping (like for a nap) then it's up to you to make the decision whether or not to let her be on her tummy. Of course it's not recommended and I certainly wouldn't do it at night but during the day maybe you could keep an eye on her?
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