Erm, the title of this thread's a bit misleading. I was expecting to see something weird about acrobatics or whatever.
In fact, as it sort of fits into the context of the thread title, I'll say here and now...no lame acrobatics like Obi-Wan. Okay?
A forward flip, backward flip, spinning around and rolling around the floor are all very well (in fact, just use Heretic II as a basis, that was great), but cartwheels and handstands? Er...no.
The other thing that I do think needs addressing is the swimming animation. If it could be like Rune, then I'd be happy. Nuff said.
Now that's out of the way (*takes deep breath*)...
Some of your ideas have already been discussed in other threads...and I wholeheartedly agree. The environment should have an effect on your character (like wind, slippery ice, cold etc.), and you should in turn be able to affect the environment to overcome certain obstacles.
Swinging on ropes or vines, cutting down trees/towers to bridge chasms, being able to catch hold of a ledge too far, would all be welcome. I wouldn't even mind seeing your character take the proper stance to push something heavy out of the way (like Lara Croft/Severance) in 3rd person...although there would be a greater argument for using Force Push or Pull.
Yes, make it as interactive and reactive as possible - that way you have to be more careful where you shoot, when you cut and where you go.