If Kim Davis doesn't want to assume the responsibilities inherent in a government job, perhaps she shouldn't have it, or taken the job in the first place. There is no persecution here. Ms. Davis is free to be a Christian, but she isn't free to impose it upon others in the performance of her government job. Inherent in accepting a government job is the responsibility to perform the job as directed. If I'm a US Marine (a government job), and say I can't kill Omar The Terrorist because I'm a Christian and 14 of my buddies get killed as a result, I get court-martialed. I am being forced to choose between my religious convictions and my job too, so would I be a "persecuted Christian"?
Your "logic" that homosexuality is a mental illness is very weak, if not outright disturbed. I can guarantee you that someone in your extended family is either secretly gay (I know they're not going to tell you), had a gay experience, and/or bi-sexual. You will never know that they have a "mental illness".
You really need to see someone, other than your minister or Bible Study teacher, because you need some help we can't give you here.