Dog radical (犭) for non-Han ethnic groups
I can only provide a partial answer:

Many of the characters used in the names of non-Han ethnic groups were originally derogatory. After the founding of the PRC, the government conceptualized New China as a multiethnic state (多民族国家), and they changed many of the characters that were perceived as derogatory. I don't know if this process started under the KMT (国民党), as you suggest, and I also don't know why no one bothered to fix the one for the Jews.

Edit: The character for the Jews is a little different from those for other ethnic groups. For one, it was not an invented character. Rather, (in modern-ish times), existing characters were picked for the transliteration. Also, although it does have the dog radical, the other meanings of the character are not negative (or related at all to animals). The only extant meanings (besides being part of 犹太) are seem (好像；如同) and still (还；仍然).