Would any inkjet still work printing b&w copies when the color inks are out?

Answer 1

It depends on the printer. My older printer would continue to print with one color empty. My new one won't.

Answer 2

For best results, make a copy of the original, color image and then convert the copy to grayscale.

That way, the printer will only use the black cartridge as it's ink source. If you try to print a color image, with missing inks, IF the printer even works, at all, it will just leave blank areas where the missing inks would have gone.