May I add my recommendations as well?
If you want closed back, fun sounding, easy to listen, I would suggest to try the Meze 99 Classics headphones.
They are easy to drive, very efficient (you can connect them to your phone), relaxed sound, the type of headphone you mostly use when you want to have fun.
Not overly analytical, the treble is not fatiguing. These are the type of headphones used for long time listening (work, travel, out and about). They are punchy tough, with the midbass pretty emphasized, so hip hop, electronic music, metal, movies, gun shots, explosions, sound really good (at least to my ears).
Being closed back, they do get warm though.
Regarding gaming, they have a good separation, but only ok imaging. I mean, if you do not game competitively they are a good choice.
If you want open back, Philips SHP9500 or Philips Fidelio X2HR might be good for your needs, but I do not have that much experience with them.
Regarding a DAC/AMP, Fiio BTR5 is surprisingly good and practical. It is small, portable bluetooth receiver dongle that can be used as an USB Dac.
Here is a review from ZReviews:
Edit: Besides what Zeos says in his review, I’ll add a few details from my own experience with the BTR5.
If you have inline microphone hadphones (with a 3.5 mm connector), the BTR5 can use the microphone and the inline remote. It also has a microphone in it’s body, which is pretty good for calls.
It gets warm if you use it as a USB Dac decoding DSD or other high bitrate music and set it to keep the battery full (it has an option to disable charging until it restarts)
It also reports battery discharging in 20% percent increments, so from 20% battery to 0% you do not get any warning and it will shut down suddenly. This is the most irritating thing about it from my point of view.