It may provide a cleaner signal depending on your mobo's onboard sound card but it's not strictly required.
Do you already use standard analog headphones with it? If so, and you don't get any hiss or weird noises coming though then you can just get an
1/8" / 3.5mm to RCA cable and connect your mobo to a dedicated amp like the Magni Heresy.
If you do get noise through it then a separate DAC
may help. It's also possible to get ground loops between the DAC, amp, and the PC that will add noise which wasn't there before.
That's a great DAC and ADC but the headphone out it pretty weak, even weaker than the Fulla. It would be great if you wanted to add a pro microphone for your voice chat or something though.
Most audiophile headphones are much less efficient than typical consumer headphones and you may not get a satisfying volume out of the Motu M2 or the Fulla with the HD800S. Don't believe the pseudo-magical claims a lot of people make about how amps alter the sound. It's mostly audiophool mythology. There two most important things in an amp are the residual noise level (so sensitive in ear 'phones don't hiss) and power output (so less sensitive headphones can go loud enough).
If your onboard sound is already clean you can go with just an amplifier like the
Schiit Magni 3 Heresy,
JDS Atom Amp+, or
Topping L30.
If your onboard audio isn't that great then a Schiit Hel or
Topping DX3 Pro+ are worth checking out.
Also, regarding the HD650 and it's variants. They are very light so as to never weigh you down and fairly open so your ears shouldn't get too hot. On the other hand they clamp tighter which may get annoying after a while and have smaller and shallower ear cups which will almost certainly press against your ears. I have both the 650 and the original 800 and for me the 800's comfort wins by a mile. Of course a lot of comfort is personal so whatever you go make sure to buy from a place with a a good return policy or used so you can resell without much of a loss
The 650's soundstage doesn't touch the 800S' either. If you need positional audio with comfort and can afford it, the 800S is excellent choice.
There are
some gamers on here, but this forum is primarily focused towards audio in the contexts of primarily hi-fi music and secondarily home theater so most people here probably weight their priorities a bit differently than what is required for competitive FPS gaming. I'm a gamer too, but in much different genres, so even I might be leaving something out.