This is NOT normal behavior on any PC. It is not a power-saving feature. If the background program is not showing on the taskbar, then it has closed prematurely.
PCs are only supposed to do what we want them to do *or* what they are programmed to do (which, by definition, is what we want them to do). If your PC is doing anything that it is not told or programmed to do, then it's either misprogrammed or under the control of someone other than you.
The point is that this PC is either under someone else's control or not working properly. I have many years of experience in this stuff and I would scan aggressively for all types of malware. If I found something bad, I would assume that there is more than the scanner found and I would reinstall Windows because I am justifiably paranoid. If it found nothing, I would assume that Windows is whacked and needs to be repaired or reinstalled anyway. Any way you look at it, the end result is a reinstall.
Because I'm paranoid, I do the right thing and back up my system regularly. If *anything* happens that I cannot fathom, I don't have to reinstall. I don't have to spend a week trying to figure it out. I restore a backup in 30 minutes and go back to work.
Try System Restore. Scan for bugs - aggressively. If no joy, consider reinstalling.