I would either stretch your budget by $20-$30 or find a pair of used Beyerdynamic DT770-Pro 80.
Comfort: Insane comfort levels. I used them for gaming for a few years and used them on 30+ hour plane rides.
Closed: You block some outside noise, and people don't hear your games.
Bass: Tons of bass for hip-hop and games!
Build Quality: You can throw them in a backpack and never have to worry about them.
Like I said, I used them for gaming for a long time until upgrading recently to DT990/600's. The DT770-Pro 80 just does such a good job for "fun" gaming with the added bass. Hip-hop/rap will sound excellent with them too. They don't "require" and amp either. It's worth looking into them.