Very nice work!
I have no idea. Lots of people seem to do it and it looks cool 
I suspect there must be *some* evidence for it, but that it was nothing like as common as we wargamers like to show it. 
There is evidence, and interestingly it points to painted helmets being something common soldiers did rather than the wealthier nobility. If you had money, it seems you paid to have your armour polished to a high sheen. If you didn't have money but still wanted it to look cool, you painted it yourself! There's definitely museum examples of sallets painted with chequer-board patterns, the actual shape and construction of the helmet itself being on the lower end of the scale. There's a good lecture Tobias Capwell on this subject which can probably be found somewhere on youtube.