Problem is solved!
Thanks all for your advice. Here's what I did.
1) Learn how to re-string a floyd rose floating bridge.
2) Find an allen wrench
3) Remove the strings.
4) Run to the store to get a new stiff toothbrush
5) Scrub every fret (with the grain!)
6) Break for dinner
7) Scrub some more
Difference now is night and day. All the white "ick" is gone. The toothbrush is black.
8) Apply some Dr. Stringfellow that I purchased earlier in the day
9) Wait a bit
10) Buff it off/in/whatever
11) Re-string
12) Enjoy
I'm totally happy with the results. The white spots are gone, the fretboard is noticeably cleaner and the new strings, as always, made a huge difference.
Lesson 1 learned: A 30-day old guitar is, in fact, a guitar that's been played by 1000 people over the course of say a year or more in a big open store. I feel like I should have disinfected it before taking it home. Ew... New guitars are only "new" if you get it from the factory.
Lesson 2 learned: "Lemon Oil" is rarely just lemon oil. I'm a believer in Dr. Stingfellow now.
Lesson 3 learned: There are about 100 different recommendations online for what is "right" and "wrong" for fretboards. I picked this one. I'll let you know if it ruins my guitar.
Lesson 4 learned: I'm an obsessive idiot. Well, I already knew that actually...
Here's a new pic: All better. The white specs are only noticeable in direct sunlight or with a flash. My arm is too tired to scrub more.
Thanks again all. I learned a lot these past few days.
GAD