Yes, that's the logic board. There's nothing soldered to it, but it is held in by a series of gouge-your-eyes-out annoying metal bumps on the right side (if you have the SE down on its face with the bottom facing you). They're a pain to get up, but you'll eventually learn the trick to 'em and it won't be so bad.
Yessir, your Macintosh will work just fine with just the floppy disk; 'tis how they originally worked in the dark days before hard disk drives. Also, if you're going to dismantle the HDD, be SURE to do it in an insanely clean and sterile environment. One stray bit of dust on the platters and it's prolly gonna be toast when you spit it up again. I also had the original HDD fail in my SE; the best option is to simply replace it.
eBay or the Trading Post are your best options. Older ones aren't all too common on eBay, but tend to go cheaply because nobody really wants a 500-meg HDD anymore
You don't really need to go for an ultra-high-capacity drive, either, because most of what you can do with these machines just doesn't take up all that much space. I've found that the 40-meg HDD I put in my SE is more than adequate for all tasks I ask of it 
Best of luck with your SE; they really are great little machines!