It depends on the setting. You have the freedom to say that rape should be legal. But if you're a teacher and say this, you'll probably get fired. You'll still have your free speech, like you could make a website about legalizing rape, but you will likely never be able to work in a school again.
All of the studies say that being gay isn't a choice and that bisexuals definitely do exist. Again, you have the freedom to believe what you want, but the rest of society has the freedom to look down on you for these beliefs.
Actually, it depends on what the mental illness is. If it's an illness that doesn't cause very irrational or violent behavior, then you could still own a gun.
Fox News, so it's probably bullshit. But either way, this doesn't mean transgender people are dangerous. At one point, Spain put people in jail for not being Catholic. Does that mean Catholics are more dangerous than people who aren't Catholic? Of course not. It was just a stupid law created by some extremist Catholics.