1) Orion, Jaron: Yes!
The "It has to be things that are sorta kinda possible but way beyond probable." sums it up.
Fighters (and monks and other nonmagic users) should be doing stuff that it is possible to imagine that a sufficiently trained human who has tapped into the Primal Power of the Human (Mind, body, soul, insert any or all *here*) could theoretically learn how to do, even if though this may be things that no lesser human by (meaning anyone on Earth, historically, ever, or nearly so) could do.
But still human. We may want Jedi like feats of acrobatics, but we probably don't want to have someone teleporting from one end of Helm's Deep to the other. We probably don't want something like the Foxtrot comic where Jason and Marcus are playing D&D and Jason, defending Helm's Deep kills the 10,000 balrogs heading at his fortress with his elf's +255 bow.
We probably don't want something like Seven Samurai, where forty brigands kill five samurai (four? I seem to recall only two surviving).
I think a problem we have here is that people want to make Fighters able to compete with the spell users...but the "So make spellcasters less powerful!" aspect is viewed with dread. A bit of both is going to be necessary. Even if fighters can shoot from one end of the Milky Way to the other, having the game mean that all 20th level characters (or all epic level characters, or all any-point-we-actually-expect-to-play-to) being able to defeat Hordes effortlessly would be terrible. Heroic fantasy is about beating hordes not simply because you're awesome, but because you know what to do and are awesome enough to do it.
If the heroes can simply wipe out hundreds of enemies without difficulty, Helm's Deep is not cool any more. That would suck.
2) I'm going to agree with Orion here, to be honest. We either need to pick a class and stick to it, or hammer out what level we're setting this craziness at and go to the class most in need of being boosted to meet that or weakened to meet that as the case might be, and make it work at that level without making it boring or too complicated.
It doesn't really matter what class, but we need to pick one of them.