Interesting. I've got 6 MS keyboards here and none say that. But that does not mean anything. I note you said PS/2 so yours is likely pretty old.
The thing to note is the ATX Form Factor standard requires all ATX compliant power supplies for PCs continue to supply +5Vsb standby voltage to several points on the motherboard whenever the PSU is plugged into the wall (and if equipped, the master power switch on the back is set to on or 1). So when you shutdown your computer, you really are just putting it in standby mode, not totally powering it off. This standby voltage is what allows us to press a keyboard key, or wiggle the mouse to wake the computer. But this voltage can also keep some possibly corrupt settings alive. So removing power completely by unplugging ensures all voltages are removed and thus ensures a total (cold) reboot.
It would seem through your #5 and #6 above that your troubleshooting steps identified the problem to your failing keyboard. This is verified by the fact a second keyboard worked fine. So it seems it is time to shop for a new keyboard.
I don't know who makes the keyboards and mice for Microsoft, but I really like them. On my two main computers here, I have the
Link Removed. The curved keyboard and built-in wrist rest are easy on my aging wrists in terms of
RSI problems. And universal (not contoured) mouse feel great. This is important for me because while right-handed, I use my mouse with my left hand.
That said, keyboards and mice are HIDs (human interface devices) and every human is a unique creature. These HIDs are extensions of our own, unique hands. As such, what feels great at the ends of my hands may feel awkward or even painful at the ends of yours. So while I can comment on the construction, reliability, key-press feel, mouse "tracking" and style, only you can decide which keyboard (and mouse) is right for you. So you really do need to find a computer store and play touchy-feely with these devices yourself. Try to position them at the same height as your current desk or keyboard tray. And remember, the store floor models have probably been abused. Oh, and take some hand-sanitizer with you!
Being a keyboard and mouse "set" is nice because they share the same USB dongle, taking up just one USB port. I have seen up to 6 of these wireless sets used in the same office and there were no problems of interference between them.
The only downside is the same with all wireless keyboards and mice - batteries. This is one area where I strongly recommend the use of rechargeable batteries. On this computer, which is used heavily for 4 - 6 hours every day, I have to replace the batteries about every 5 months.