About the 10th or 11th century.
True, but not of a size that would allow the painting of a complete lifesize head on glass. Such panes were extremely expensive. Glass panes were blown... and then flattened against a large plate of metal... introducing variations in thickness and distortion.
I don't think there's any merit to this theory. If the shroud is a fake, it's faked through some method of transferring the image to the shroud of a sculpture or dead body. It might be fake, but the current claims supposedly proving it false have a lot of holes in them. Like I said, I'm neutral on it, but I do find it fascinating.