Well, Alduin is the only dragon who actually is seen eating anything. And even then, he just ate souls. Alduin is on a different level than other dragons though, seeing as he's essentially a god. I speculate that the dragons feed on travelers near their nests, or perhaps they hunt animals. That would explain why you can find dragons out in the wild at random. I always assumed they were hunting.
Oh, dragons can definitely eat stuff. It's their kill-move, kinda like decapitating. It's happened to me before when a dragon was much stronger than me for some reason and I was low on health.
They don't swallow you in that animation. Though we can make the assumption that Mirmulnir did eat some of the guards at the watchtower, as a set of Whiterun guard armor can be found on his corpse.
They definitely eat things the first dragon you fight and kill always has guards armor and **** on him, you may not see them eating the guards per say but it is clearly implied that other dragons eat.