My dog named 'Bravo' is playful and clever.
Shouldn't it be Bravo (without quote marks)?
I wouldn't say it's absolutely wrong, but it's certainly not necessary to use them. Some people might be slightly more inclined to use the quotation marks if the name of the animal were an actual word, which bravo is. However, they're not needed. In fact, the vast majority of people would just say "My dog Bravo is playful and clever."