Well, the E6 is not a DAC if that's what you're planning. It only amplifies the signal put out by your soundcard, so if it's bad already it's going to still be bad - only louder and more powerful. ;)
What I first wanted to suggest is the FiiO E10 or Alpen (depending on your budget, the Alpen is more expensive). However, those DAC/Amp-combos cannot put out 5.1 sound signals. It's "only" stereo, so you wouldn't get to you use your 5.1-system. However, there are soundcards able to run 5.1 configurations (don't ask me which ones to look at, I'm not knowledgable about them). So all in all:
1) No, sorry, to my knowledge the affordable ones are not 5.1. There might be some DAC that I have overlooked (logic says there has to be one) that is able to put out 5.1 but I don't know about it, so, can't help you there.
2) The E6 is a good portable amp (but not a dac). The bigger brother to it would be E11. But I don't think any of those headphones you've been looking at need an E11. Go with the E6 if you want an affordable portable amp to connect to your laptop and mp3 player.
3) Yes. Most laptop soundcards are not that great in my experience, in that they're not able to drive headphones fully. The E6 would help in that aspect. As for your logitech sound system, probably not - I don't know what it could do for your system just by amplifying the signal. I doubt your logitech system needs (additional) amplification of the signal between your laptop and the onboard amp on the speakers.
I hope I've helped a bit.
What headphones did you come to the conclusion would help you in your quest for audio nirvana?