Amazon Kindle supports Korean text (for models starting with the K3, which is all sold currently) and Amazon has an excellent reputation for rock-solid firmware stability. When it comes to Kindles, it "just works".
Paperwhite has WiFi (with a more expensive option that includes 3G), frontlit display, with adjustable brightness, and weighs 7.3 oz. so it's pretty light. The Kindle Store includes tons of free books automatically generated from Project Gutenberg. All models can be easily sideloaded.
It can read .txt, but not EPUB. However, the format it supports instead, AZW3, is just as good as EPUB, and all your EPUB's can be easily converted to AZW3, using
calibre.
You WILL want to use calibre anyway, as this is the only way to customize every aspect of the metadata of all your books, and in bulk. Fine-grained control can be done using plugboards to, say, derive a Title that includes at the end "(series_name, Book x)" on the device, based off of the title and series metadata fields.
A word of warning: You do NOT want the Nook SimpleTouch with Glowlight, unless you wish to have 3/4 of the internal memory walled off and accessible only for official B&N books downloaded from the store. They don't like sideloading, as apparently it cuts into sales even more than the hatred this engenders in would-be customers.
And they removed the MicroSD slot for the same reason.