Congrats on your 558 purchase. They're a fantastic little pair of headpones, with a nice upfront midrange to keep things interesting, as you've already noticed. Not so sure about your headache problems - I've never found my 558s fatiguing. Then again, I am a Grado owner, so maybe I'm more tolerant of forward upper-midranges than some others. I use an E7 to drive my 558s, and it's a great cheap amp/DAC for them. All the hissing from your internal sound should be gone, and things might sound a bit more balanced than your internal sound due to the E7s low output impedance (it's possible you've got great internal sound, but from your description of it hissing, it sounds unlikely). As for the ground loop problem, I do have a friend who still has mains hum on her USB headset, so I'm not 100% sure it will solve that problem. Still, the E7 is such a flexible tool that if you were already prepared to spend that money, I wouldn't hesitate too much about it.
If you're looking for an adaptor for 1/4 to 3.5mm, I would take a look at the Grado one (google Grado adaptor). It's a bit more expensive than some others, but it's got a short bit of wire between the two ends that alleviates the stress a very long jack/adaptor combo can put on the port.