Headphone Recommenations

Not a headphone, but the portable BTR3 from Fiio is pretty good, lets you use most headphones as BT.

I would not buy headphones with Bluetooth, and if I did I wouldn't spend that much. Bluetooth headphones are more for convenience than audio quality. The Sony Sony WH-1000XM3 seem to be well regarded.

Perhaps you can elaborate on why Bluetooth is a must have? Most members here are focused on sound quality and two channel stereo systems with speakers and not as much on headphones. You might have better luck on headfi or Reddit.

A great pair of headphones in your price range are the massdrop Focal Elex. No Bluetooth though. 

Via BT, the sound quality is very much determined by the codec used, as Bluetooth is a lossy transmission method. The best SQ seems to come with the LDAC codec, as in the Sony headphones mentioned by @big_greg. I have a pair of the immediately preceding Sony model and find them good value and good sound, a touch warm perhaps, but very comfortable and enjoyable. I listen to a wide variety of music, like you.

Noise-canceling technology interferes with sound quality, IMO, far less than using less capable BT codecs like aptX. One might even say that NC improves the sound by raising the signal-to-noise ratio. As audiophiles are fond of saying, "It lowers the nose floor." So in the presence of anything from traffic noise to a humming refrigerator, I’d rather listen through NC headphones than far more costly non-NC headphones. Need I say, not everyone agrees.

P.S. Though the Sonys connect using LDAC automatically from an Android device, I am not sure whether LDAC is easy to get using Windows, iOS, OSX, etc.

Good luck!