If you're on a budget, it's hard to beat the D7100 for stills and you still get excellent video too. I have taken some pictures with my D7100 that shocked the crap out of me with their sharpness and clarity. I've used macro lenses with other cameras and I *thought* I knew what sharp was, until I got the D7100. In my opinion, it's THE best bang for the buck right now. The D5300 could enter that category when the price drops a bit ($700 or less) and the D5200 is still pretty good, but you won't get images nearly as sharp for photography, but you may see just a smidge more DR. In terms of video, they all three should be identical. I can't tell any difference between the video of my D5200 and D7100, I would imagine the D5300 will be identical to those two quality-wise, but with the addition of the 60p 1080 video, which I'd never (or VERY rarely) use anyway. If I were going to buy the D5300, I would wait for someone to do a moire/aliasing test, but I don't anticipate that it will be a problem, since it supposedly has the same sensor as the D7100.
If you need low light, I'd say 5D3 or D800. Samuel's Flaat Shootout 2012 video showed that the 5N made an excellent showing, too, but it wouldn't quite be in the same class as 5D3 or D800.
As far as autofocus with video, none of the Nikon cameras are even close to ready for primetime. I've tried lens after lens, different settings, different situations, it doesn't matter, autofocus just sucks for video. For stills, the D7100 is brilliant and focuses ridiculously fast and it's always spot-on. So really, if you're only doing video, you won't notice any difference between 5200, 5300 or 7100 because you'll be manually focusing anyway (the 5200 and 5300 lack the focusing motor so it only works with "G" lenses, you will be stuck manually focusing with "D" lenses). Of course the D lenses give you aperture control (manual) in live view, so that's why I go with those when I can. The only time I can trick the D7100 for photography autofocusing is when I'm shooting a strongly backlit subject and they're moving toward/away from the camera, but even that isn't a deal-breaker, for me anyway.
If you need low light, I'd say 5D3 or D800. Samuel's Flaat Shootout 2012 video showed that the 5N made an excellent showing, too, but it wouldn't quite be in the same class as 5D3 or D800.
As far as autofocus with video, none of the Nikon cameras are even close to ready for primetime. I've tried lens after lens, different settings, different situations, it doesn't matter, autofocus just sucks for video. For stills, the D7100 is brilliant and focuses ridiculously fast and it's always spot-on. So really, if you're only doing video, you won't notice any difference between 5200, 5300 or 7100 because you'll be manually focusing anyway (the 5200 and 5300 lack the focusing motor so it only works with "G" lenses, you will be stuck manually focusing with "D" lenses). Of course the D lenses give you aperture control (manual) in live view, so that's why I go with those when I can. The only time I can trick the D7100 for photography autofocusing is when I'm shooting a strongly backlit subject and they're moving toward/away from the camera, but even that isn't a deal-breaker, for me anyway.