It might not have the power it once did (if it ever did, I don't know) and it's used very loosely in a jocular way most of the time, but yeah I'd say it's mildly racist. I don't know many Jews who'd take kindly to being called Hymie..
No we certainly do not! "Hymie" and "Kike" are about as close as one can get in the U.S. to making an N-word level slur against a Jewish-American. I don't know that "Mick" could should be considered in the same category as either of those words, but I agree that it is still mildly racist (albeit less powerful than it once was). It would be more like calling someone a "Yid." If used seriously in a derogatory manner, it might be too anachronistic to really offend someone. However, I wouldn't recommend referring to people as such, even in a jocular manner.