How do I keep CPU off when charging? When I connect the USB power source, it makes CPU to stay on (even though "Development->Stay awake" is off).

When charging from a weak power source if makes battery to drain even faster than is case of USB power is not connected at all (because of CPU stays on instead of sleeping).

How do I stop external power connectedness affecting CPU sleeping policy?

I'm not talking about screen being on, it's about CPU being awake (unfortunately it is not easy to see if CPU is on or off).
Long press on the power off switch while plugged in, next tap on the power off prompt.   When it powers off make sure the charging light remains on.    If light is not on press power on quickly just so that the battery percentage shows in the center, it should blank screen again and the charging light come on and remain on.  You can retest battery charge percent again anytime.  When ready to use  long press on power on/off to on and watch or wait for rebooting to take place.