Discrimination against eye color would definitely be forbidden as discrimination against race; you can't ban everybody except those with blue eyes without having a racially disparate impact. This is probably true of those other physical characteristics too.
But speaking more generally, the principle in America is that if something isn't forbidden, then it is permitted. You could probably get away with some nonsense like "Don't hire anybody who owns a red car, because red car owners are too impulsive and hot headed"
Yes, and by the same token, discrimination on the basis of caste would likely be judged to have a disparate impact based on nationality, race, and religion (against Hindus / Indians), since such discrimination won't impact people of other races. We just don't have actual precedent about it yet, but any company that doesn't take action over it is playing with fire.
Yeah, I can't think of anywhere in the US this would overlap with a protected class - so long as it was consistently applied. If someone says they discriminate based on your sign, it would warrant a very close look to be sure they aren't actually only checking the birthdates of people of certain protected categories.. this sort of red herring discrimination does happen, but if it was truly and consistently based on someone's sign, that would likely be defensible everywhere in the US.
But speaking more generally, the principle in America is that if something isn't forbidden, then it is permitted. You could probably get away with some nonsense like "Don't hire anybody who owns a red car, because red car owners are too impulsive and hot headed"