In that price range you are not going to find what you want and cover all the bases. Your best bet is to get at least 2 cameras. What you are looking for is a Swiss army knife and what you are going to end up with is
that huge one that is so unwieldy its unusable. If you refactor your thought process and opt for two cameras instead I will say,
In your case you can snag a nice used D3300 to cover your static video and photography needs, pair it with the kit lens or the 35 1.8 or something. It has a mic input so you can capture better audio than you could from some internal mic. This is a great camera and will shoot video well above any quality you actually need.
Then take the rest of the money and buy a Used older GoPro (one of the ones that wil do 60fps @ 1080) and use that for small hand held and action shots.
I will also make a not on shooting high resolution and high frame rate. A lot of people these days consume content on their phones, iPad's or not in full screen on their computer. It frankly does not matter if you shoot the original footage at 8K and 120FPS is the guy watching it is streaming over 4G on his iPhone 5S....
Regards
Dave