So I was wondering if I should get Dawnguard out of the way as early as possible or wait on the main quests till I complete the Skyrim MQ and the Civil War till around level 40?

It depends. I personally like to finish the MQ before I start any of the other quests, but with Dawnguard there's the problem of the vampires attacking villages as soon as you have it installed. This can be really annoying as they may kill non-essential but quest-related NPCs. This stops after you finish the DG questline. At least, it did for me. So if you don't have DG installed yet, I'd wait with that until you get further into the main game. Then again, Dawnguard gives you access to some pretty neat weapons and armour from the start, such as the crossbow.

Btw, does anyone know how it'd work with dragonshouts you learn during DG's storyline if you haven't mastered the power of the voice yet?

You mean without Dragonborn Story? I don't know. You still have to unlock them I think.

The same way as shouts in vanilla game before you start the MQ - you can learn them, but not unlock them.