(Many) Questions about Kayle

Where do I start.. A year ago, I used to build Nashor's Tooth and Lich Bane and proceed to Carry. Nowadays, I feel like I don't know what my role is. I've been losing , I think all of my Kayle games.

1. What lane do I go as Kayle?
2. When are her power spikes??
3. What do I do in teamfights? Seriously though, I feel like I don't do any damage, and just sit around waiting to ult. I can't get close because I'll get blown up. Is she played like top/mid lane Lulu?
4. In what order do I max her abilities? I know Kayle is a champion who is really diverse, so please give your reasoning too.
5. Hybrid, Off-tank, or Ap build?

Thanks for taking the time to answer !

Edit..

It is actually better to max Q W E, Runaan and Berserkers is actually very good on her. Since you max E last.