I've seen some passively cooled D510 boards which seem just right.
There's only one known-good board with fairly low power consumption, the Intel.
I should probably check he PSU in the case
It's probably not very efficient.
Otherwise, what characteristics should I be looking for, or avoiding, if my goal is to get an efficient non-brick PSU for the new motherboard?
Same as with hard drives: look at the power consumption in reviews with reliable meaurements such as SPCR's.
The one non-brick PSU which compares with a pico with a good brick is the Kingwin/Superflower Platinum 550W. It's probably way too expensive for your purpose. I think you'd be better served by a brick/pico type of solution.
And big file transfers is my usual use- reading/writing 200MB+ photoshop files back and forth for example.
You should be limited by your drive's speed if you use a single cheap and low-power drive for storage.
If you want to speed up transfers for a limited number of files, there are tweaks you could use depending on your exact usage.
A lot of people seem hail the Intel based LAN controller on the Supermicro motherboard vs the Realtek controller on the D525MW. Do you think there is any real world benefit in a case like mine for going with the Intel based NIC?
In order to take advantage of the better NIC, you'd need a fast drive. And affordable fast drives consume more power, even when idling. The Supermicro board itself is expensive and would consume extra power.
So if you want that speed, you're going to be spending extra money and consuming enough power that one has to wonder if it wouldn't be a better idea to build on a faster, more expensive and less power-efficient platform. Atoms have I/O limitations by design and lack server features like VM extensions and ECC support. They're great for small and relatively slow servers but you're only going to push them so far.
The only reason people should consider the Supermicro board in my opinion is if they want the brand or the IPMI feature.
And considering what you've lived with until now, I don't think you have a use for something better than a basic Atom anyway.
If you want low power consumption, invest your money in a pico and a good brick. That, an Intel D5xx board and a drive selected for its low power consumption would give you an efficient yet affordable little server. You could do better with the new Atoms though.