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.