Thanks Hadron for the update, I could not try it before since I'm still not rooted.
Anyway, from the app description, it looks like you can set the bot up at will:
"PowerBoot is a streamlined application that lets you control when your Android smartphone or tablet boots. Through its on-screen interface you can boot your device with scheduling control over the date, hour and minute of the boot."
A bit misleading I'd say, but of course I trust your try on this.
On your second point. Well, I'm not a hardware expert, but I think every OS can in principle (even if definitely not in "friendly-mode") be set to boot at a specific time. You can with Linux for example (and I guess also with iOS,Win), so why not with Android.
The point is, even when you shut the device off, the internal clock is still ticking otherwise time would be wrong at next boot up.
You "simply" have to add something like "when clocks ticks this time -> do X". You might be not able to set X as exactly whatever you want, but I'd expect that at least boot up could be an option?
It's probably difficult from the coding point of view, but why shouldn't it be doable?