#1
So I bought a DD-3 for my gear, played around with it, got used to it a little. Got myself so I knew where I could use it, and how I should use it effectively. Leave it in my basement with my gear, and someone knocks over a bucket of f*&#ing paint on it. Now the pedal won't even stay activated when I engage it. Any words of advice, things to check for/replace, or am I screwed here?
#2
Probably not as the solvents in the paint would probably destroy the electronics in it. Using paint thinners or other products to remove the paint will do even more harm.
#3
Open it up and take a look; best case scenario the paint just screwed up the switch (doesn't seem likely if it initially engages then turns off) or is clogging one of the switch contacts so that the pedal "thinks" you're unplugging your guitar.

Even if the problem is more serious, it can't hurt to open the enclosure and look. Do you get sound when the pedal is bypassed? When you engage it, does the effect start for a little bit before stopping?

if it's not fixable, it's your basement. It shouldn't be too hard to figure out who spilled the paint right?