Firstly, the Nikon cameras just use one mount, the so-called Nikon F mount. You can mount any sort of lens to a DX camera. I usually buy lenses from eBay, so that’s where I checked. I cannot talk about weather-proofing because I have no experience with it. I keep my stuff out of the rain. Maybe think about Buying a cloak in olive green or something that blends in with the natural environment to throw over yourself because it will cloak you and protect the camera and the lens from rain.
- Nikorr Af-S 200-500mm 5.6 ED VR ( AF-S in Nikon speak means the autofocus is internal, so it will be nice and snappy.)
- Nikkor AF-S 70-200 2.8 FL ED VR ( Used it will still be over budget for you, but the quality is absolutely worth it.)
- Tamron 70-200 2.8 DI VC G2 ( Basically the above lenses little brother.)
I think one of those three should meet your criteria.
I like the crop factor because it essentially makes your optics better for free. Since it will only use the center of the lens, which is its best part. From your experience, does a speed booster actually have a large impact on optical quality, since you are adding an additional glass element?