I am tried of being attacked by the Dawnguard whenever my character goes out into the countryside. They attack even though my character is cured of vampirism. So I want to do this quest. I've finished the main DG quest line with the vampires and killed Harkon. I've reaquired vampirism (thanks Serana).
But if I talk to Garan or Fura, they just tell me that Feran Sadri has something for me and he gives me radiant quests. I've done a few, but still not getting the Destroy quest option.
1) what do I have to do to be given the Destroy the DG quest?
2) if I destroy the DG, will attacks decrease?

Keep on doing those radiant quests, you'll eventually get it.

But even after you destroy the Dawnguard, you will still get random attacks in the countryside, they will just not be as frequent. They will be maybe be a bit more frequent than the random vampire encounters.

IT took me awhile but I eventually got it. Fura Bloodmouth gave it to me.

i like how gamesas implemented destroy the DG and destroy the DB but no destroy the thieves guild or the companions. I would gladly pay $20 for a dlc that i can kick both these factions out of skyrim forever.

You better watch out...we have contacts in Windhelm that can easily see you rotting in a dungeon til the next era

But how are you going to contact those contacts when no one in your guild ever leaves the Ragged Flagon?