The first photo is of mineral deposits left behind when you boiled your water away. If you water is really hard (full of minerals) you may get these even in normal use. Barkeeper's Friend cleaner and some hard work can remove it.
The second photo is showing that you actually managed to clean the dark oxidation off some of the bottom, but not all of it. When copper is really clean (free of oxidation) it's quite bright and pinkish or orange. The way your pot was before, it had a fairly even layer of oxidation, which makes the copper brown. You could either clean it all the way--again, barkeeper's friend is a good bet--or leave it and it will soon be brown again.
The last one I'm just guessing about. I think it is probably mineral deposits too, if there was water underneath the pot, or if if boiled over. I do not know if something like Barkeeper's Friend would be appropriate for your cooker, though. It could also be oxidation or staining from the pot--something on the pot that burned off and left a residue or discolored the surface. It's also possible that the hot pot actually damaged the surface, in which case the discoloration may be permanent.
In any case, your pot is not damaged if the bottom is still flat (sometimes overheated pots warp). Cleaning it will make it look nicer, but none of the things you're seeing are likely to have an impact on how it cooks.