The HD 600 is the better can (I am not a fan of the HE 400's mids). However they are not $200 better. See: http://www.head-fi.org/t/634201/battle-of-the-flagships-58-headphones-compared-update-audeze-lcd-2-revision-2-6-4-13#user_HD600 and http://www.head-fi.org/t/634201/battle-of-the-flagships-58-headphones-compared-update-audeze-lcd-2-revision-2-6-4-13#user_HE400 Pick your amp/dac after you pick your cans--and don't cheap out. Select one that works especially well...
You--especially you who has placed comfort above all else--need to try these cans before you decide.
Headphone comfort is a very individual thing. Sizing and sensitivities are unique to each person. We can point you to headphones that are generally considered to be comfortable and mention others that have garnered complaints for being uncomfortable. However, the only way to know for sure is to wear them yourself for a time equivalent to your longest listening...
K 712 over HD 600 due to soundstage.
The DT 880 is too bright for you.
All that said--there are many excellent cans in this group and many have quite different sound signatures. You would be doing yourself a favor by listening to as many as you can before deciding.
Your Pioneer receiver probably isn't old enough to have a decent headphone out section. The industry started chiseling on that sometime in the mid to late 80's. Give it it a try and see how it sounds. I only have enough information to guess--and that's not much good. Hopefully your receiver has a tone out button that will disable the eq feature.
Don't count on much improvement from the amp. The Grados are very efficient and can easily be powered by any portable device. If the DAC in that device is decent, you should not need an amp. Any improvement is likely to be slight. The quality of the DAC is more important.
For the "i" series the 225 was a significant improvement and considered by many to be the sweet spot of the SR line. However, I have not yet heard anything above the 80 in the new "e" series, so I can't help on it.
You are more or less in a dead zone in that price range--even in the U.S. market. Assuming you can get U.S. pricing closeout pricing on the UE 6000, it is a great buy for under $100. Absent a sale or closeout, there's not much that's significantly better until you get to the $250 to $350 range. That's even more true without an amp. The KRK KNS 8400 is very good, but benefits from an amp.
The e is different. It delivers more bass than than the i version. However, there is no guarantee that, without listening first, that you will like the 80e better than you MS-1 (i?). There is even less of a chance that you will like it much better.
It tells me that their probably better at marketing than at engineering. They present a nearly bear claim of what their product can do (and presumably do well). Without reliable third party confirmation, take it to heart at you own hazard.