Reruns of Tom and Jerry were a favourite cartoon while we were growing up. However, when I watch those cartoons again, I question myself whether I would allow my children to see the same again because of the violence and racist elements.
No, it is not too violent. We watched it as kids and it did us no harm. As for the race issues, it is a product of its time, but it would be wrong to pretend it never existed. First of all it would go over most children't heads, and if it didn't then you simply explain to them to context of when it was made. If you're still concerned, there is a censored version of Tom & Jerry where Mammy Two Shoes was edited out and replaced with a white teenage nanny. If you ask me, that's worse, because it is taking black people off the screen.