Should I unplug the power when the battery is 100% or not?

Most lithium ion batteries, like those in your game controllers, should be unplugged when they reach maximum capacity.

The battery technology is ideal when the battery is between 20 and 80% full. Keeping the battery on the charger when it is at 100% can negatively impact the overall lifetime of the battery’s ability to hold a charge.

Yes, that is the normal procedure. The only time the console wouldn't power off is if there was a fault with the controller, or when the battery is coming to the end of it's life cycle. When it's life cycle is ending, it may no longer register a full charge,
in which case the console may not turn off.

I would just unplug the Play & Charge from the controller to be 100% sure.

My battery indicator shows up 100% of the time. Have you tried a reboot after you installed the update? Try this, with the console on, press and hold the power button on the console for 5 seconds until it shuts off. Unplug the power supply, wait about
30 seconds and then plug it in and reboot. You should see the green Xbox One splash screen upon reboot. It may take a minute to boot all the way to the home screen.

Controllers with a built in battery usually have a cut off when the charge is 100%, so it should be fine when connected with a cable.

[quote user="IceStorm III"]You do still have the battery pack in the back of the controller, though, correct? Even when empty the pack should be on the controller if you're using the charging cable.[/quote]

I keep battery clip in pad, but no batteries, because it increase weight to pad, and after longer play I'm getting tired

Why wouldn't you try unplugging the power cable and then reattaching it? That would force the controllers to sync up again with the console.

When I unplug cable from controller and plug it again pad is not recognised by console (manual synch not work either)

You mean when you power the controllers from a PC USB port, they work fine with the 360 for extended periods of time?

When I use X360 pad to play games on my PC they work perfectly fine, problem occurs only on 360

It sounds like your 360 is having trouble keeping the controllers powered continuously. Perhaps invest in a set of rechargeable batteries and a charger? Then you wouldn't need to power them via the P&C cable.

I have rechargeable AA batteries but problem didn't disappear