I think Mixamp's have just the one output for headset?
Assuming that is correct, then you'd need an adaptor cable to join headphone and microphone cables into one 3.5mm jack.
This is the type of cable you'd need:-
https://www.ebay.co.uk/itm/3-5mm-He...659148&hash=item4b4c146549:g:u84AAOSwI2xbdS39
As for the headphones; best choice for FPS gaming is AKG K702.
They are more power demanding than other headphones though. Whether the Mixamp can drive them loud enough, I'm not sure about as Mixamp's are really intended for use with headsets that are much easier to drive, such as Astro's own. I suppose it depends on how loud you like the volume.
Sennheiser HD598 are another good headphone for gaming. Not quite as good as the K702, but they are considered to be easy to drive headphones, that pretty much anything with a headphone socket can drive. Just to give you an idea; K702 need 4x as much power for the same SPL volume.
Beyerdynamic DT990 are another good choice. More demanding than the HD598, but not as much as the K702.
If you can still get hold a pair of Philips Fidelio X2HR, they are another decent choice. Easy to drive, like the HD598 and unlike the others, they take a straight 3.5mm cable input, meaning you can use a V-Moda boom pro microphone with ease and have just the one cable for headphone and microphone, as you would with a headset.
If you didn't already have the Mixamp, Creative G6 would be a better device as it has a much better headphone amp and SBX prostudio, which is better than Dolby Headphone that Astro use. £120 is a lot to shell out though, when you've already got something that can do the job for a number of headphones.
As for microphone: V-Moda would ideal for the X2HR, otherwise a Antlion Modmic would be best choice for the other headphones. Modmic can attach to most headphones, but you do end up with two separate cables, which you would need the above adaptor for. Some people do tie the cables together for a neater solution.
Using the V-Moda with the other headphones is either impossible or difficult. DT990 has a fixed cable, so can't be used with the M-Voda. K702 cable is detachable, but AKG use a mini XLR connector. You can get a mini XLR to 3.5mm connector, but they aren't common and add bulk and distance between microphone and headphones. HD598 uses a 2.5mm twist lock design, so you'd need some kind of adaptor. Again, that would add bulk and distance from microphone to headphones.
