I can't remember exactly level wise, but I'll do my best. Let me preface this post by saying the only DLC I've played thus far is the Pirate's one. I'm waiting for the level cap increase before playing the other two.
I think the Pirate's DLC starts around level 20ish. The Hammerlock one is around 30, and I can't remember Torgue's. These are the minimum levels you should attempt them, otherwise you'd have a hard time.
As for story progression wise, the only one that assumes you've beaten the main story is Hammerlock's. You should definitely finish the main story before doing that DLC, as the story in the DLC will spoil the ending if you don't.
Hope this helped, sorry I can't give more info. I'm waiting to get back into the game in a few months once I can gain exp again!
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