I wanted to come back and close the loop on this after talking to Supermicro a couple days ago.
First of all, the rep was telling me that if a motherboard and a chassis were meant to be compatible it would say so on the right hand side of the web page, and neither this board nor chassis mention compatibility with the other one. I understood this general rule of course, but I pressed him about the I/O shield accessory for this chassis that specifically calls out the model family for my Atom board. In the end he agreed that the board must work in this chassis as long as I buy that shield, even if it doesn't say so on the website. Really no better answer than the assumption I'd already made, but I took it and moved on.
On the power supply question, we went round and round and he put me on hold twice to go ask someone else about it. I could tell I'd finally gotten through to him after I explained why I felt there might be a "missing piece" to convert from the 12V DC barrel jack to the 4-pin ATX plug on my motherboard. Rightly or wrongly, I explained to him that I assumed his chassis takes the 12V barrel plug and converts it to an 8-pin ATX connector for his Xeon-D motherboard. That may not be correct, and he wasn't sure either, but the point was, I'd need an adapter from 8-pin to 4-pin in that scenario to plug into my Atom board. Since that adapter surely didn't come with the I/O shield, how would I get it? This was when he put me on hold the second time, and when he came back he was confident he had the answer. What he said is that when you buy the power supply accessory kit (84W?), it comes with the adapters you need to convert from the barrel jack input to the different ATX connectors. I didn't really want to buy their power supply, but at least this seems to answer the question.
For the final question I ended up not asking Supermicro, both because I'd been on the phone with them for over 45 minutes by this point, and because I believe I found the answer in another person's SuperServer build thread. What it looks like is that the chassis and super server come with the physical bracket to mount an expansion card, but not the riser card you need to make the electrical connection.
So with all of that information, it does seem like you could buy the E300 chassis and use it with an Atom motherboard, as long as you buy the optional I/O shield and buy the optional PSU kit (or else rig up your own with adapters from Amazon). If you want to use the expansion slot, you'll either need to buy the right Supermicro riser card, or you might be able to get by with a generic flexible one from Amazon, and just use the included bracket to hold the card.
I ended up deciding that the odds of me upgrading with a 10GbE NIC down the road are pretty slim, and bought a trusty m350 chassis with a few fans, extra HD bracket and power kit. I still like the idea of the Supermicro chassis, but at this time it's just a little pricey and harder to get all the special pieces together to make it work with Atom.