Doesn't seem like I can jump. I got to the part right after the intro, you grab the skeleton key and go through the big door. Right after that you have to jump over the pit. She says, "If only I could turn into a swarm and fly at will," or something like that. I assume that means mist form, which when you fall, you fly back up. But mist form doesn't do that for me, I just hover. Also, I can't jump. In the controls it says Jump is bound to space, but that's only a dodge.. I checked the walkthrough, and in that version, they're jumping just fine. I don't understand what's happening lmao

You need to stealth the weapon to jump.  "In Combat - space is dodge"  If you push the Z or x button I think is to sheath weapon.

Press R to unequip you weapon, then you can jump =)