M350 Case - 71$
PicoPSU 80W - 52$
All your prices are high (Polish taxes?) but those prices are really excessive.
If you want to use a board which needs a regular power supply, get a case which includes a fanless power supply and you'll save money (especially since they include a brick and you wouldn't need to pay so much for shipping).
will get lower performance than D2700
You'll also get lower temperatures.
You don't need more performance than a N2800 for that kind of server. Nor do you need 4G of RAM by the way.
About AC/DC brick - what should I buy?
If you don't care about a few watts of power consumption, you can reuse an old brick.
If you buy from a reputable vendor, any brick with the right specifications will be fine.
But if the Seasonic is not too expensive and you know it's compatible, you might as well buy one instead of wasting your time looking for an alternative.
And about electricity in Poland - 1kWh costs ~0.086$
In that case the cheapest solution might be to reuse old gear no one wants.
I think it makes more financial sense to sell your AM2 than to use it for this project. But I assume you want to keep using your old hardware regardless...
If you want case/PSU advice you should explain what you mean by "silent".
If you buy an Atom board and a M350, no explanation is needed: the HD will be the only noisy part (be careful though: some laptop drives are quite noisy!).
But it would be expensive to make regular desktop hardware literally silent. So you should explain how much noise is OK.
And by the way - what do you think of using one of these fanless mainboards without chasis at all? I know that more dust will cover it, but I will save extra $$$
Dust isn't a real issue in most environments (and the M350 wouldn't protect the board from dust... or from liquid spills for that matter) but the chassis protects the board against a number of threats.
If you can put the board where nothing will accidentally fall on it or simply be pushed against it and where and no children or animals will reach it, I don't think you need a chassis. You could also build a rudimentary chassis yourself, perhaps by cutting and folding and a wire mesh and putting a sheet of non-conductive material like balsa under the board. People have also made rudimentary cases out of thick paper but that's bad for fanless cooling.
Running without a chassis would get you lower temperatures by the way.