Of course they do. Nightmare AI has a significant Money/Xp/Warscore Boost. Think its like +30% so whilst you have earned 10k, they have earned 30k. Think of all the items you can get with 20k...
As far as I can tell the AI (the actual intelligence) is the same in each mode, its just how much of a handicap is given.
It's not only that they have more money, so they buy a bunch of artefacts and go slaughtering. For the movement speed i'm pretty sure that this much is not achievable by normal means(as i wrote: UB with +40% movement speed was clearly outruned by rook slowed down by about 20%). Take down a citadel and look how your movement/attacking looks. The attack speed and regeneration is in theory possible, unfortunately i can't tell if they have any artifacts. Point is that they start killing only after this triggers, until then they usually have near zero kills, so their only income are gold mines and creeps.
EDIT: Also it doesn't happen every time. I have been playing against nightmare AI since very beginning of beta and i have seen this happen only about three times. So i don't cry that my ass got kicked(in fact, we have won the last match, but only by holding all portals, the AI was invincible).
