The m50x and Hd 558 are completely different sounding pairs of headphones.
The m50x is closed back, more rugged and portable, has pleather ear pads, and has a greater clamping force. It provides accurate, detailed, and powerful audio. Listening to it you can tell these were made to monitor but it still works well for listening to music. For gaming, not so much. Especially if you game for long periods the clamping force can become unbearable and the earpads will get hot and sweaty. Also the soft pleather earpads harden over time so keep that in mind.
The HD 558s are open back, use velour earpads (very comfortable), lightweight, and will also have a much wider soundstage which makes it the better choice for gaming. However since it's open back its bass won't be as powerful and punchy as the m50x. Sound wise it is smooth, neutral,and natural.
Keep in mind the pros and cons of closed vs open back headphones.
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An AMP would allow you to drive the headphones to higher volumes with less distortion. It shouldn't be needed since both of these headphones are low impedance. A soundcard acts as a DAC which you may benefit from depending on whether or not it will be an upgrade from your onboard DAC. Also some soundcards have surround sound software that might be worth looking into.
For up to $350 you have some really great choices available, but I recommend going out and trying a few high end headphones before buying them to find the sound signature which best suites you.