No one is more badass than Ryu Hayabusa. No one.
As was said before, Ryu is just trolling in DOA. It would only take one inferno ninpo, piercing void etc. to annihilate any of them. Ryu is canonically much more powerful than Ayane, or really anyone for that matter.
No. Most of us like to pretend that doesn't exist since it's lame and non-canon.
Momiji is obviously a good deal weaker than Ryu. Ayane deliberately waits in NG for Ryu to come through and do everything, plausibly because she knows she would be unlikely to succeed on her own (and doesn't even risk helping Ryu fight, for that matter, even when he's been cursed twice). Hayate, as you mentioned, is also weaker, and after Dimensions I'm quite convinced that Kasumi is drastically weaker also. In the DOA universe, there really isn't anyone "close to Ryu's level."
In conclusion: in DOA land, Ryu is the top of the food chain. If you want to speculate on cross-universe battles, Kratos and Dante could be a challenging fight. Ultimately, I'd give it to Dante, who posses the speed/agility of Ryu and the strength of Kratos.