From your username, I first have to ask - do these also have to fit during a full moon?
Your second concern - the Mic, is worth a lot of thought. I’m not a gamer but I used to live online at night, with VOIP and Skype meetings with production teams in India. Also did WebEx training. So needed to think about voice quality. I still go to occasional conferences and meetings, and am always concerned about sound quality when communicating. @TylersEclectic and a few others do game and can tell you what works for that.
When I want quality sound on Skype these days, I use my Mac Mini and have a Blue Snowball USB microphone.
https://www.bluedesigns.com/products/snowball/
These are available (base model, iCE ) for about $50 at the usual suspect stores. (Guitar Center, Best Buy)
The sound is WAY better than anything on a headphone. They would work fine with your laptop. As it’s an actual real Microphone, you don’t have to wear headphones to use it! Much better if you want to record video, and even pretty good for Skype with (gasp) speakers instead of headphones. Great with IEMs.
For plugging into a telephone, I find the 1MORE line has good inline microphones. I see you have a set of 1MORE piston… my EDC (Every Day Carry) are the Triple Drivers.
At my office, I use a Plantronics SAVI 7000 series headset - an older DECT model similar to the 8200 series now.
https://www.plantronics.com/us/en/category/office-worker
DECT is the standard used for wireless phones, and quality is excellent for voice.
I don’t think that any of the wireless headphones have quality that is as good for music/classical as wired, although some are getting better. Wireless phones are limited both by the amount of power they can draw from battery, and by the technology making the wireless connection. There is a lot of discussion here and elsewhere on the forum.
I have a pair of Sennheiser MM-450s that are OK. They are wireless or wired (noise reduction only in wireless mode). I get about 4-6 hours on a charge. They are completely unremarkable and unimpressive. They are considerably older than your HD 4.40 selection, but were also about 2.5x the cost new.
My guess is that you may end up with 2 rigs - one for gaming and one for music. The music may end up being wired.