The Amish certainly do have disabled children. I’d also like to point out that the Amish have no issues with modern medicine and will go to the hospital if that’s necessary. Certainly, there is no prohibition on Amish women giving birth in hospitals. A lot of the communities will make an exception to the “no electricity” rule for life-saving medical equipment and powered wheelchairs for those who need them (each community makes its own rules, so there is some variation between Amish groups).
As noted, the Amish have a limited gene pool and thus sometimes have children with some very rare diseases. The Amish don’t practice arranged marriage, though, with the awareness that marrying close relations is not a good thing, there is some effort made to meet people from other Amish communities. They are allowed to use trains and buses to do this, so they aren’t restricted just to how far you can go in a horse or buggy.
Outsiders have, at times, converted to being Amish but it’s rare.
Whether or not the Amish will in the future make more use of genealogy and genetic knowledge remains to be seen. Since it’s a matter of health (and sometimes life or death) they might decide to adopt more testing technology.
The Amish do keep to themselves, but they aren’t nearly as isolated as people sometimes think. The Amish just to the east of where I live do interact with the rest of the world, just in a manner they control. Back when I worked in the Chicago Loop there was a group of Indiana Amish that would come into Chicago for farmer’s markets and to sell their farm products to the workers downtown. As I, too, live in Indiana I sometimes found myself on the same train they were riding. They can be quite charming and personable, and are aware of things like major current events (they do, after all, read newspapers). There’s no prohibition on them talking to “English” (that is, the non-Amish), doing business with outsiders, working for outsiders, or hiring outsiders (the Amish going into Chicago hired someone with a van to drive their goods into the city for them). It’s just at the end of the work day in the outside world they go back into their communities. You can also bump into them shopping at Wal-Mart or Home Depot… while their dress, hairstyles, and customs may be different than what the rest of us do in my interactions with them they come off as pretty normal people and don’t seem to have anything to hide. (Of course, some of them do - they have people struggling with drug addiction, mental illness, and so forth just like everybody else)
I don’t know how that compares with the FLDS crowd, I have never met one so far as I know.