With regard to entering a zero, the correct button to use IS the 0/10 button, (aka key 10 - in bold).
I have successfully set up codes using it - if the LED does not blink OFF during code setting when pressing this button then the button does not work.

Obviously the button can be tested by selecting channel 0 on TV, to name but one. Make sure the button is pressed firmly.

Yes, it does appear under other brand names and I think Mitsubishi had little to do with the manufacture; probably made in China.

JW