You could be willing to try a more affordable brand like the Floureon BM-800 @$30. This is what I had been using and have been quiet happy with it myself.

The wire doesn't matter. The HD599's are not USB and will be 1/4 headphone jack or 3.5 mm ( like your phone has). The HD 599's don't need an amp but unless you have really good onboard audio, you will want to buy a DAC or DAC amp combo or a soundcard ( they have good dac's on them and usually have decent amps built in). Most people just get a DAC amp combo.

If you have the usual realtek onboard audio you will notice a difference with a proper DAC. You can just buy the HD599's and plug and play, they will work fine but if the quality doesn't blow you away then you need to get the DAC. The sound should really be much clearer than your steelseries, i mean like laughing when you listen to music and saying "wow". At least that is everyone's reaction to switching from gaming headsets to sennhiesers and other proper headphones by real audio companies.

You Don't need to EQ the 599's but it isn't a bad idea since they don't have really strong bass ( its not bad at all though).

They don't have a mic so you will want to buy one. I got these:

https://www.amazon.com/gp/product/B00FFIGYOI/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

https://www.amazon.com/gp/product/B01KPKCJ2G/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

I ended up getting them on sale for 35 bucks for both. You might can find a really good deal too.

Otherwise a lot of people get modmic, which doesn't sound quite as good and costs around the same price as the full condensor mic setup like I bought.