I don't think a single word would work for you in both cases, as they're different contexts.
The different colors implies a variety of colors is what's pretty, not a color that's not the same as the current one.
They have a different dog isn't implying a variety of dogs, but a dog that isn't the same as the current one.
So, I think for the "different colors" where you're trying to emphasize variety, you want something along the lines of いろんな or 種々{しゅじゅ}の, so:
「種々の色はきれい」
「色んな色はきれい」
Or, just to be annoying:
「色々の色はきれい」
:)
All three of these mean something along the lines of "various colors are pretty" or "different colors are pretty," which is slightly different than saying "the variety of colors is pretty:"
色の種類{しゅるい}がきれい
For "a different dog," you'd probably want to use 別の over 違う as it sounds more natural to my (admittedly gaijin) ear:
「ううん、別{べつ}の犬{いぬ}を飼{か}ってる」
As an aside, your use of 持ってる implies they're literally holding a dog in their hands, not that they have or are keeping (飼う) a dog as a pet.
「ううん、違う犬が持っている。」
<-- This means "The other dog / Another dog has (it)". "They have a different dog" would be 「違う犬を [飼]{か}っている」. You know「を」 is the object marker, right?