I am looking for a quality and durable headphone with high audio quality on a budget

Durability whise I can recommend Beyerdynamics, they should have some good (wired) options at your budget like the DT 770. Or the 990 pro

I think you really have to consider whether you want wireless or not, because wired will give you much better sounding options

You can also take a look around this website, they are quite thorough https://www.rtings.com/headphones/reviews/best/music

Sennheiser is pretty good; I got a wireless pair from them. Main problem for me is that they get sweaty really easily, so I can't do more than just sit at my computer while wearing them. Their prices also appeared to have gone up, too.

A bit less than $200, but the Audio-Technica ATH M50Xs are a durable and quality workhorse. I've been using mine going on 10 years now and the only thing I've needed to do is replace the ear pads and add a protector around the head cushion. Highly recommend. Plus it has detachable cables of varying lengths and coils.

Side note: they do fold up for portability.

If you're willing to hit up Ebay for "new in box" options, you can get some pretty good stuff under $180. For wireless:

Fairbuds XL: Made by the Fairphone folks, who have a very good reputation for repairability and longevity.

Sonos Ace: Stainless steel pivots, replaceable earpads, huge battery, no glitchy touch controls. Mine have treated me well so far. Absurd you can find them this cheap, considering the original MSRP.

I use the Sony MDR-7506 headphones with the Fosi Audio P3 Tube Preamp which together come to around $200. The headphones on their own are $98 and work very well with sound quality that for me seemed better than the slightly more expensive Audio Technica M40x head phones I had been using for years prior. Combined with the preamp the MDR-7506 headphones really take on a rich sound with the added benefit of adjustable Bass/Treble and bluetooth connectivity. I also really like that the band on the Sony headphones is made of metal, and the hinges (they do fold) are sturdy since I have a big head and have had problems with other ones breaking over time.

Sennheiser HD 6xx. Hands down the best for what you're looking for aside from portability. Although their build quality is amazing and you can toss them around without worrying you'll break them under normal use. They're so comfortable I almost prefer not to take them off.

For the last 15 years I've bought 3 pairs of headphones from AKG and I'm extremely happy with them (why 3 pairs, because the first one was stolen, I gifted the second one to my dad and I am still using the last one)

They are extremely durable, especially if you pick a model with removable cable. I've used the K702 almost every days for the last 6 years and it barely shows any sign of wear.

I personally love the way they sound but it's not for every one. They are more built for professionals with a very neutral sound but very precise.

Does anyone else rely on the media controls on headphones? I feel like once you add that requirement the options are a short list. I checked out a few mentioned here but none seem to have media controls. What does everyone else do when they listen to podcasts or something?

I've had the same bose quietcomfort headphones for at least 8 years now. I've replaced the ear pads a couple of times over the years. They were on every day during my commute before I was WFH, almost constantly at the office so I could concentrate, and now indoors every weekday and outdoors several days a week where I do farm work. Still no issues and great battery life. If you can find new old stock or used but well-cared-for, that would be great.

Are you looking for wired or wireless headphones? If wired ones, check out beyerdynamics DT770. Very common model for studio use, spare parts are widely available. Not that portable though. There are different models (pro, pro x) with slightly different features. One offers a detachable cable for example.

A more portable alternative would be Sennheisers HD25, which are THE DJ Headphones. As that, they usually get thrown around a lot but deal with that well. Spare parts are also widely available.

For wireless ones I can only think of the Fairbuds XL, which are constructed in a way to be very easily repairable, including changing the battery.

I have had my paid of Sony wh1000xm2 for about 7-8 years now. They still sound incredible but the leather is peeling a bit on the replaceable ear covers, I have never got around to it cause they are still comfy. Some of the best noise cancelation ever, wireless and wired operation and controls on the headphones.

Essentially I am saying, a good option would be picking a headphone in that line. Honestly any newer generation is likely to still be incredible and durable.
Just recently I had seen the 5th generation of those headphones on sale with a pair of wireless earbuds as a combo for around $200 and you can get just the headphones for cheaper than that. Or just spend $40-50 and get an old pair of wh1000xm2 and be happy.

I can recommend against getting the Shure SE215s durability wise, mine went back to the factory twice for cable/connector/monitor issues.

I can also say the same about my Isotunes free aware bluetooth ones. They died about a month out of warranty.

Sound wise, I prefer the foam ear tips vs the silicone because I listen in a loud environment. They are both pretty good in that regard. The Isotunes tips fall off after a couple months and need replacement though. They are both noise isolating and the free aware has mics that can allow limited sound through but I turn that feature off because I have not found it to be a quality implementation at all.

I occasionally play bass guitar and keyboards and have found Behringer's studio-monitoring range of headphones provide excellent quality reproduction for around AU$50 - better than many AU$200 consumer-focused brands. Sennheisers are also good but more expensive.