Edited by frostmine - 10/31/16 at 8:54pm
They're most likely being way under-driven, especially if you're hearing scratchy bass. Bass usually suffers first from being underpowered. I know these headphones are rated at 100 ohm, and certainly not intended for portable usage.
Yeah, definitely. A fiio e5 doesn't do it justice.
Exactly how "thin" are you looking for? I also have a set of the Beyer DT770 - 32 ohm version. It can be powered relatively well out of an iPod alone (since it's only 32 ohms), so an E5 would easily output the necessary power to drive them fully. In my listening experience, they can keep up pretty well with even my QP400s. Definitely have got a lot of bass and are closed, perfect for a portable headphone. They are still a little bulky though...when in the case that Beyer sends with them, I can only fit the cans, my laptop, and maybe one or two other things in my backpack with me. They're quite a bit more expensive than the insane $60 you got the 220s for, but I really like them and it was worth the $193 I paid at B&H.
The DT770 sounds like something that might be up your alley, but I could be totally off on that.
Unfortunately the QP 220 is probably the least efficient of recent MB Quart phones, and is the model least likely to sound good through a media player or laptop. Even the Fiio E5 won't be up to it. It's not really a reflection of their quality; they're just designed to demand more power. They'll sound better through a home stereo's headphone jack, or a desktop headphone amp.
Incidentally, the "3 lbs" in Sonicelectronix description is the shipping weight, which includes wires, adaptors, display box, padding, and shipping box. The actual phones are less than half a pound, not including the cord.
Phones in your price range tend to have idiosyncratic sound (it's kind of shame the Quarts aren't working with your equipment, because the Quart house sound tends to be pretty evenly balanced), so make a memo to yourself with what you like in terms of sound, and then start searching head-fi to see what models of headphone pop up often in relation to it.
I don't have any helpful recommendations, sorry. If you have to stay within the $100 bracket, you will probably do better by watching the for-sale forums here and get some decent bang for your buck with a little more seller accountability than you can expect on Ebay.
The advertised specs say it's 63 ohms (which is moderate impedance for headphones) and 106dB (efficient for full-sized phones). It's going to be easier for your iPod and Fiio to drive than the QP 220.
I can't guarantee this will sound good because I haven't tried them with a setup like yours, but in my experience with other full-size Sony cans, the sound isolation will be decent, and that means you'll be able to get good audible sound at lower volumes in noisy environments, which also helps with the listening experience.