How common is it for insurance to pay for genetic testing if I want to get it done for a hereditary birth disease?

Check the fine print. It depends on the insurance carrier. Some conditions are common enough that they are covered. An example is maternal screening for down's syndrome (trisomy 21).
Pre-approval. Many insurance companies require pre-approval of genetic testing. They also may not even consider such unless you have been seen by someone with a geneticist in their group.