Very rarely. If you have an ally close to an enemy, you can sometimes force them into war with your enemy. I think most times the ally will just cancel the alliance, but not always.

As far as when to DOW over just marching in, possibly DOW could be considered more polite by the AI. That is, you may be able to secure peace with them easier later on, and the other nations might see you in a better light. I don't know if this is actually in the game but it seems logical.