Honestly I don't think that a mental stability test would do anything for someone with a mental problem trying to become a member of a social network. The reason why it wouldn't work is that they would probably lie on the test, which would just make the testing useless. Second, you have to remember not to give specific information about yourself like actually posting your actual picture on the social network site and most importantly do not give the city you live in. Everything that we tell our kids not do should also be done by us adults as well and you should have nothing to worry about.
Hope that helps.