first: stop creating a whole bunch of new threads for the same question. you have this one thread, use it and wait for an answer patiently. multiple threads will be merged here or deleted.
--------------------------
how powerful is your onboard audio?
that question is often not an easy one to answer given actual output is generally not stated like it would be on a dedicated amplifier. taking a quick look at the motherboard in question it looks like the audio section is about average so i would assume output to be more mediocre and average than anything special. on average onboard audio can power easy to drive headphones good enough but may be found lacking for anything harder to drive.
modmic uni vs omni
the omni will pick up sound from every direction, the uni will pick up from one direction. if you have ambient room sounds the uni is what you want but the omni is a bit more sensitive i believe.
m50x
v-shaped signature, good bass levels without muddying treble too much, folding design, very small soundstage, fairly small earcups.
dt770
similar to m50x but better all around. v-shaped, strong treble and bass response, decent soundstage for closed can, big earcups. can be ran without an amp in some situations but for best bass and best sound at minimum a decent soundcard or cheap amp would be suggested.
373d
basically a new version of the 363d which is known for being bass light with decent soundstage. it is rather expensive so it is questionable as to if it is good value
game zero
similar to the 373d, it is bass light. this could be good for gaming for some advantage but if you want a more full sound and good bass levels it may not suit your tastes.
game one
based on the hd518/558 headphones. neutral sound signature with decent soundstage. a decent all rounder.
hyper x cloud
a decent choice for a budget headset but it really is nothing special when compared to headphones in the 100-200 euro class.
ad500x
not mentioned by you, but this is a well known bass light headphone with very large soundstage. quite popular for gaming but if you want that "fun" sounding audio with good bass this is not the can for you. it is however rather advantageous for things like fps gaming.
---
not sure what country you are from, but the dt770 or dt990 seem to average around 139,00eu and the modmic for around 66,00eu. that would be about 205,00eu for just those two. you should be able to find some form of amplifier or the creative z for 70,00eu or under.
the game zero looks to be around 200,00eu and the game one around 160,00eu. the hd518 around 100,00eu. not sure of country but i did see what looked like a sale on the hd598 on amazon fr for about 130,00. normally they go for 180,00-200,000 as at/senn products are generally expensive in europe.
the m50x looks to be around 160,00eu
the ad500x looks to be about 120,00eu.
i'm not intimately familiar with the eu market but beyerdynamic prices generally are cheaper than audiotechnica, akg, sennheiser and many other brands which makes them a little more easier to afford. why, i dont know.
if you want something which is easy to power then one of the choices by audiotechnica or sennheiser like the hd518/558/598, ad500x/700x or game one should be okay. you might not get the "best experience possible" with onboard but i am fairly confident they will at least work. the game zero should as well depending on model as one older model was a bit harder to drive but i believe the newer models are not. the hxc is not hard to power.
as far as amplifiers, on the cheap i generally recommend fiio products but on the eu market it is hard to find them. the creative z as mentioned earlier has a decent headphone amp included. otherwise there are certainly headphone amps in your market but you might need to do some research to find out what is good and what is not if you go that route.