No. Toodles has a large black maid who is only shown from the neck down. But there's no particular reason to believe this is the same character who (depending on the film) either owns Tom or runs the household in which Tom lives.
Jerry, as usual. He instigates a rivalry between Tom and Butch for Toodles. But in the end, despite his size and his species, he wins her from the two cats.
Yes. It's included as part of the VHS compilation Tom and Jerry's Cartoon Cavalcade, which shows it uncensored. However, it has not yet been released on DVD.
The name of that tune is "Way Down Upon the Swanee River". It was also used on a few other MGM cartoons as an accompanying score to scenes of cartoon characters in blackface.