iDevices actually have some of the most linear output available for a portable device, it is just that the overall output is just not that strong; 50 Ohms is pushing it. That said, what is the primary use for your HD598, indoor or portable? Are you content with using your iPad or prefer your PC?
You will definitely benefit from an external DAC/amp. Like other have said, onboard audio is terrible (I have the ALC898) despite the claims by the manufacturer; i.e. S/N ratio, THD etc, are examples of what looks good on paper... Anyway, if you have a home theatre/stereo receiver the HD598 are an excellent match. Something like the Audioengine D1/D3 or the FiiOs (what is with Head-fi's obsession with them?) you mention will also be of benefit to the HD598. If you have the older iPad with the 30pin connector go with the E17 as it supports 24bit/96K output (and includes the interconnect cable), otherwise you will need to purchase the Camera Connection adaptor for the lightning connector. The Schit Magni and Modi combo are also reasonably priced in relation to what was mentioned and is something you should consider if you are willing/able to purchase a DAC/amp in the $150-$200 range.
I will like to add that yes, the better the DAC and amp, the more the HD598 will reflect their quality. Going from ALC898 to my D1 wasn't a huge leap overall, rather it provided a cleaner, smoother sound. Going from Toslink in to the D1 and then RCA out to my Yamaha receiver yielded the same gain as going from the ALC898 to the D1; nothing huge but it did improve upon the 100Hz impedance spike and smoothed out the bass (and gave it more kick). Going from the latter to my Teac UD-H01 was like getting a different pair of headphones all together.
I think, realistically, I'll only ever use these indoors. I would prefer using them with the PC as it's all HDMI'd up to the TV (movies/games etc).
I'm quite happy with the output from the iPad - I think the dac/amp would stay with the PC. Is there any particular sound difference between the E10K and E17? It's a little daunting starting down this audiophile business. After all, the headphones did only cost 74,99€ ($93usd~). I'm pretty sure I'm not maximizing their potential though - argh. Wallet at the door, indeed.
Thanks for your advice and apologies for my limited vocabularly when it comes to the audio world
If you're happy, just stick with what you've got for now then upgrade later. A DAC such as the HRT Microstreamer can be run from your computer, ipad or anything else for that matter, so is fine for at home and portable use. It sounds amazing and is one of the best upgrades I've ever made, but I didn't miss it until I got it!