There are no "real" 5.1/7.1 headphones, those are pure marketing garbage.
High accuracy two channel headphones are all what's needed.
Human hears in 3D because head&outer ears cause direction dependant changes to signal received by both ears from sound source.
http://berkeleybside.com/sound-dimensions-of-binaural-recording/
Besides dummy head recording that can be simulated by mathematical algorithms.
(that's feature most sound cards have had for 10+ years)
And if headphones are accurate enough besides directionality there's even sense of distances.
Closed design is just huge challenge/drag for that needed accuracy.
So naturally most gaming stuff is closed, making them anything else than actually capable for good gaming immersion no matter the signal.
(millions of flies can't be wrong: poop is good...)
Actual audio makers don't do much of wireless stuff.
Sennheiser RS-serie is something I would trust to have good sound, inside limits of cup design and has good battery life. (up ot 18h)
And they use easy to replace AAA rechargeables instead of some at worst glued on its place proprietary battery.
With model updates Sennheiser has just moved price of open design model upwards in price with RS185 costing toward £300.
While without wireless absolute top level headphones for competive gaming/detail can be bought for not that much over £100
But having wireless microphone is something those don't have, or those are mostly aimed at communication purposes instead of headphone sound quality.
Meaning it's pretty much only gaming trinkets which have wireless microphone.
On the other hand you likely don't need be able to use mic from anywhere in the room, so some desk mic could work.
