I'm not a Nikon shooter, but I know Canon has the option to turn on Auto ISO, so I'd definitely check to make sure that's *not* on. If the camera ups the ISO to compensate for a dark part of the framing, you'll end up with noise you don't want. Just a thought

If $1000 is your max budget, I'd say make sure you keep support gear in mind. A decent monopod or tripod runs a couple hundred bucks, and is invaluable.
Also, none of these cameras are going to ship with a decent fast prime, but mirrorless cameras give you a ton of options for adapting old manual lenses (FD, M42, etc.) - I've got a $10 Minolta 50mm1.7 that's pretty wonderful.
I'll second the audio recorder - I personally use the DR-40 mentioned above, but the Zoom H4N is pretty standard, and it's nice to be used to running gear that everybody's used to.
Nady has a ~$35 shotgun that, while far from perfect, is going to get you better sound than an on-camera mic.
Also, lights and stands are pretty essential. There are some decent LED panels for under $50 that can throw a little light, and of course, a trip to a hardware store can get you something that will work in a pinch.
Extra batteries! Whatever you're shooting with, have spares. There's not much worse than running out of juice with one last scene to shoot.
As far as cameras, I haven't used, but have heard good things about Sony's mirrorless APS-C cameras. The a5000 is only about $300 right now with a lens, and it might make a good starter camera.
Good luck