Headphone Recommenations

Another option you might consider is a closed back headphone to not bother the rest of your family.  It won't keep all of the sound in the headphones, but will be much less of a distraction to others.  I have a pair of Audeze LCD-XC (no bluetooth) that I really like, but those are a little out of your price range.  I also really like the Massdrop x Fostex T-X0.  I think those were about $150 (no bluetooth).  The T-X0 sounded really bad at first, but after 50 or so hours of break in it opened up.  It's still a "dark" headphone, with more emphasis on bass than treble, but it's still revealing and very easy to listen to and comfortable to wear.  It appears to currently be sold out on Massdrop (now called Drop.com), but you might be able to find some used ones.

Keep in mind that for most headphones you can get longer cables or extension cables. 

For your stated use Bluetooth might be OK.  I listen to Tidal, Qobuz, Spotify, Radio Paradise, etc. in my car via Bluetooth and the sound quality is pretty decent, but a car is a much different listening environment than sitting in your living room with headphones on. 

Audeze Mobius right at $400 new.
Bluetooth and wired so you have options.
They tout and sell them as a "gaming" headset which is a little off the mark imho.
Not the lightest but comfortable enough for a couple hour sessions.

Still breaking mine in but suitably impressed so far.

Went to can jam in New York this winter to specifacly audition over the ear Bluetooth headphones. I tried them all and ended up getting the sennheiser pxc 550. For me the best combination of sound, comfort, and incredible long battery life. The also are corded.  I have 7 pairs of over ear cans and about a dozen different ear buds. Sound signatures for all of them are different so u really just need to put em on your head and let your ears decide!