I don't have enough insight into this to give a great answer, but in general the Amish use more home remedies and alternative medicine than the average American. But they're not opposed to doctors and hospitals when needed, that's always an individual choice.

They do both home births and hospital births, but I couldn't say how the percentage breaks down. My sense is that home births are less than half in the more progressive Amish communities, and more than half in the more conservative communities. But I don't have any good data to back that up.

I'd say it's similar to childhood vaccinations, where I often hear the claim that "the Amish don't vaccinate their children, that's why they don't have x problem." But that's simply false, many do vaccinate. Again, it's the parents' choice whether to do so or not.