Yes, if you want a no-muss, no fuss experience for books, K3 will get you there.
Other brands work fine, too, and even offer features Kindle doesn't but you also have to roll up your sleeves and do some work. Just to make that clear.
Going through the list:
Yes, fanfic is usually DRM-free so Kindles can read it; HTML and PDF natively, most everything else via conversion. Calibre will be your friend. That Fanfic downloader tool you linked can already output Kindle-ready mobi-format, so you should be covered with it.
And yes, the price differential for the larger KDXG is noticeable; given the paucity of shipping competitors (none, today) apparently there isn't much volume production of the screens and component cost tends to be inversely proportional to the volume. Volume sales seems to be in the six inch range so...
Anyway, the K3 with G3 will preferentially try to use the WiFi for its wireless transfers and, if it can't make a connection, *then* switch to the G3. So the choice there is whether you want 24x7 connectivity everywhere or are happy with the WiFi-only. Just remember that the connectivity is for transfers and the *experimental* browser, it *will* work fine with the wireless shut down (gives better battery life, too).
Sounds like you're covered.
Make sure to check out the "recipes" list in Calibre; those are websites that it can automatically download and convert to ebook form for you. You can write your own "recipe" but they have a *lot* already compiled and ready to use.
By your taste in fanfic it sounds like you might have an interest in Urban Fantasy and (maybe? SF&Fantasy), if so you might want to check out Tanya Huff's Victory Nelson series. And check out the free ebooks at these web sites:
Fifth Imperium - CD images of promo CDs with dozens of DRM-free SF&F ebooks; entire series in some cases.
http://baencd.thefifthimperium.com/
BAEN Free Library - Free ebooks chosen to introduce the works and worlds of some of the best SF&F writers out there. Lots of goodies. Check out MOUNTAINS OF MOURNING, for one.
http://www.baen.com/library/
Webscriptions - Is the ebook store associated with BAEN and a few others. DRM-free, sub-paperback pricing, no geographical restrictions, multiple format support, repeat downloading in any format. They sell you the *book* not the file. Plus they also sell their namesake monthly webscription bundles of 6-7 books for US$15 instead of US$6 per book. A great way to discover new authors and genres. In addition to BAEN they also carry NIGHTSHADE books which tends a bit more towards the urban fantasy side of the spectrum and they have their own (quarterly?) bundles. Check it out.
http://www.webscription.net/default.aspx
The Closed Circle - Three top-flight writers got tired of their backlist languishing at the hands of the publishing houses so they took matters in their own hands and set up their own BAEN-like operation. The idea is spreading.
http://www.closed-circle.net/WhereItsAt/
Hope, you have fun with your new toy.