Is it possible/practical to install and run Linux on a USB flash drive?

Yes, Linux on a USB flash drive is doable. @David Spillett discussed some of the options on another question today. The other question is Ubuntu-specific, but the general concepts ("LiveCD"-type installation versus "hard-drive"-type installation) are pretty much distribution-agnostic.

Frankly, I'm not a fan of the Intel Atom D510 CPU. Check the Atom spec chart. The Atom 330 is practically the same CPU with 2/3rds the power consumption. You can find a variety of Mini-ITX Atom 330 boards (on Newegg or elsewhere) that provide more SATA ports or other configurations. I grabbed Zotac's IONITX-F-E Atom+ION board (3 SATA ports, Wifi PCIe) a few months ago and have been very pleased with it.

I was building a desktop/HTPC though; if I were aiming for a network storage server, I'd be looking for more SATA ports. The 2 SATA ports on your D510 board is pretty limiting. Newegg lists a couple other Zotac boards with 4 SATA ports, and the NM10-B-E with 6 SATA ports. (The NM10-B-E is a Mini-DTX board, so it may not fit a Mini-ITX case. It uses the D510 CPU.)

[*] I've only linked to Zotac boards because I have direct experience with one particular model and feel comfortable recommending it. There are other Atom boards from other manufacturers; research before you buy. My main point is that the board you've mentioned may not be the best choice.