Are dragons shouts magic?

Dragon shouts are indeed magic, and thus subject to both magic resistance and spell absorb. Except for the elemental breaths and storm call (which are subject to appropriate elemental resistance,) all of them are completely blocked by Ward spells. Yes, even (and especially) Fus Ro Dah.

So you can do vampire necromage + atronach + atronach stone, or Breton racial + atronach stone, and be completely immune to shouts.

Does it make me immune to Disarm?

Probably, yes.
If past games are any indication then anything the game considers a magic effect is affected.
Same as with the necromage perk in synergy with being undead as a vampire.

Another way to become immune to Disarm is to reach level 30.