I'm pretty sure that it won't wake up from shutdown. I haven't tried hibernation. Still, I'd rather plug/unplug the cable every time instead of waiting for the computer to come out of hibernation or boot, open the applications again, etc. Sleep mode is great: back on in under 5 seconds, everything exactly as it was.
A possible solution would be to use an add-on NIC, I think I have one somewhere. But it annoys me that I don't know what the hell is causing it. By now I'd say a faulty board, but exactly the same happened with a previous, different board. I've run out of ideas.
I'm trying to refrain from celebrating until it's been a few days with no wakes, but I think I FINALLY found the answer. I'll document it in case some other poor soul has this problem and googles it.
In the BIOS menu there's an option to display the Intel ME prompt during boot. It's together with the options to display F2 to enter setup, F10 to enter boot menu, and so on. I enabled it just to see what it was. Upon restart, sure enough, right at the end of the POST there's a prompt to press CTRL+P to enter Intel ME. I did. Apparently it's kind of an extension to the BIOS menus. Among the few items inside, there was a "remote wake up" thing which was set to Enabled. I set it to Disabled. Now the computer doesn't reply to pings when it's sleeping. As I said, it will be a couple of days until I can be sure that I've solved the problem, but it does look promising.
I have to say, this Intel ME thing (I think it's Management Engine) is not documented anywhere, nobody seems to know about it, and why Intel put those options there instead of in the BIOS menus with the rest, is beyond me. Even the prompt to enter Intel ME is disabled by default, I was lucky to stumble upon it after trying everything.
Anyway, I hope it works permanently. I would like to thank everyone for your suggestions, especially rseiler. I may have ended solving the problem by myself (hopefully!), but I appreciate the time you took to contribute. I'll let you know if there are any news, or I'll mark the question as answered in a couple of days.
Good, and no doubt that's it.
Its settings are detailed in this document, which ironically I regularly consult, but I'd trained myself to ignore anything in the ME section on the theory that it's N/A. And it is for me, with a "k" processor that doesn't have the Intel vPro stuff (which includes ME), but yours clearly is not.
Worrying about the same settings in two places is a strange concept to me, however. It's not very intuitive of Intel.
I've always had this with my DG45 at home, and while I was sure it must be some hardware setting, not OS (it was the same under Linux as in Windows7), but could not do anything about it, so I had to shut down the computer every time instead of sleep mode.
Now I tried another Google search if someone has found the solution - and after years with this problem, finally there is, thanks!
(yes, I also had no clue at all there's a feature like "Intel ME" hidden in the BIOS...)