Thanks for your all support and opinions. I've found solution for my problem, however, I don't know if it will work with you. Resetting PRAM or SMC was not helpful.
First, I had to remove all the additional software for all my mouse/keyboard devices, that includes also original Logitech drivers and apps. Then, I used a terminal command to remove my keyboard type recognition wizard data, to let him run again.
sudo rm /Library/Preferences/com.apple.keyboardtype.plist
Then I restarted Mac and when it restarted - I closed keyboard recognition wizard (this one where you press keys near Shift). And it worked. 🙂 Don't let him recognize, because he will mess it up again.
Afterall, you can install Logitech Options for your mouse and to use Logitech Unifying.
(It fixed Logitech K400 Plus issues, but not Microsoft WirelessKeyboard 2000)