Going the ModMic route is definitely a great choice!
I wouldn't pay too close attention to the 'gaming' marketing, good headphones are good headphones. I use a set of HiFiMan HE-400s that I bought for listening to music and they're awesome in games too! I'd spend some time reading written reviews from a reputable source that reviews audiophile equipment, pay specially attention to the way they describe the sound signature. Also think about fit: On ear? over ear? etc. Bottom line if you buy a set of well reviewed headphones at this price point your going to be happy, focus on what will fit you taste. It's a lot of money take you time researching!
Something to consider with open back headphones is that the offer absolutely no noise insulation you'll be able to hear everything around you and others may be able to hear what you're listening to. That said I used my very open headphones all through college sometimes sharing a room and it was totally fine.
At about 100 pounds you're starting to approach the threshold of where a DAC/AMP may be worth the extra money; you will likely notice a difference (depending on how good your ear is). I'd take a look at the Fiio A3 (30 USD). even an entry level DAC/AMP combo like this will make a significant difference.
A word of warning! once you get some good audio gear you can never go back! enjoy!