Grado - SR125e

No contest koss kph40i. 40 bucks. Personally, grado sr60x for 100. If you're willing to spend a few days looking, used sr125e's for like 50 bucks. The diminishing returns on audio happens quick past about 400 bucks. 250 on iems

Real talk? I think my grado 125e's are up there with the best. I also really like the fiio Ft1 pro. Very good for gaming.

You want too many things out of one device. Don't get one thing that does all of those shitty. Get some rosetechnics ceramics x earbuds for jogging and outdoor use. 30 bucks, put on the rock eq in the app (and set the language to English) and add some $10 foam ear tips - the generic ones made for the xm4 buds are the best. They have pretty good anc with a wine noise mode that works great. Surprisingly great sound quality. No sweaty head. If you need wireless for gaming, your only choice is the audeze Maxwell if you care about good audio. Bluetooth has some inherent lag and Bluetooth microphones are universally bad. You need a dongle. And the only option that is worth the price at any price is the Maxwell. But if you can go wired, the world's your oyster. Also remember that all wireless things are disposable. They will not last more than 4 or 5 years tops. I don't like throwing away expensive things. Some good wired headphones will last forever. Plus device switching on wireless is a pain (the dongle on the Maxwells makes it better, they can be paired to your phone via Bluetooth with the LDAC codec (if you're android, if you're apple you're screwed for wireless quality but fiio is supposedly working on a tiny Bluetooth adapter that fixes that) and then just use the dongle to plug into various devices. Personally, I game with fiio Ft1 pro's, my grado 125e's or my ga3000e's. I'd I want a closed back experience, I use the apevoix Grit iems, which are only around 220 but punch well into the 700-800 range and have astoundingly good gaming technical performance too.

New? Ft1 pro, the regular Ft1 is good but not "big ass easy to drive planar magnetic that come with both 3.5 and 4.4 cables. For about 200 But year on year, used grado sr60, 80, 125, and 225's. Used pristine 125e's are like 50 bucks everywhere and they're just a fabulous set of cans, and durable as hell.. But new, its the Ft1 pro, no contest. Or iems. Moondrop kadenz are just astonishing for 200 bucks. But the simgot em6l are also amazing for 100.

Grado SR series. They're open-back. You can hear everything.

Im an audio guy and after many years of optimizing playback solutions imo there are two very clear answers: Sennheiser 6XX for full size, and Etymotic ER2XR for IEMs.  The sennies can be found new for around 200 and the Etymotics are around 130 new. I found them uncomfortable and the cables are definitely microphonic but you can get aftermarket tips. For pure $/sound i think they are the strongest I've ever personally heard.  Another outside option is Grado SRs, for under 100 I love the sound. Build quality and comfort is meh but they look cool and sound excellent. They aren't a great match for every genre though, read up if interested. 

I have some budget headphones, grado open back and m-audio closed back. I love the grados for almost everything (especially vocals!) and find them non fatiguing and super lightweight so easy to wear for a long time. But, I often want more bass and/or privacy so then I'll pull out the closed back. OP, why not get one of each and still stay in your budget?

I'm a big fan of Sony. The WF 500 earbuds are a very solid budget option, have used mine constantly for well over 3 years and still hold a charge well. And the 1000 xm5 are great over ear noise cancelling headphones that have decent aftermarket support, like cooling ear pads. I love my Grado open back headphones too, but theyre more for dedicated stereo music listening, and don't think they offer anything Bluetooth anyway.

For rock music, there is nothing else like grado, try to find a secondhand one to fit your budget better, but you really need to try it on your own at least once before buying anything else. It's open back, which mean it will leak sound everywhere and much more suitable for home usage. It also doesn't sound that good for most other genre, but man they do the meanest guitar shred I've ever heard