My friend has a problem with the PC we just built: he says that the screen doesn't turn on, but the on-board LED and fans do. However he also says that the mouse that he connects to the back of the board does not turn on. I guess it's a power problem but I have to investigate more.
If it is an optical mouse with an LED in it, than it should light up as soon it gets power from the USB port.
The motherboard is doing a POST (Power on self test) after you push the power button, normally it also checks the USB ports if they are working, after the Post, the OS should boot (or your PC is writing "missing operating system") and all USB ports should be powered up.
So yes, your mouse should turn on.
I would check the PSU, and maybe replace it, sounds like its dead.
UPDATE: I went to my friend's place with a mobo speaker and it did a beep corresponding to a memory problem. Also the red led on the MemOK! button (it's an asus board) was solid red. As it turns out the RAM sticks were not properly connected and a bit of pressure fixed all the issues.
So apparently the board doesn't give power to the USB until POST is complete, even though it seems strange.