Eggs – What would cause the albumen of a hard boiled egg to turn purple

I've never adjusted the length of time based on number of eggs. As long as the water is boiling I don't think it would take any longer to cook a dozen than to cook one (it might take the water longer to come up to boiling, I guess).

For hard boiled I normally bring them up to the boil and then turn the heat off, and leave them for 15 minutes. If you like them less well-cooked, you could reduce this to 12 minutes. Then empty the pan and re-fill it cold water a couple of times to cool them down quickly. Leave rest of them sitting in the cold water while you prepare each one.

Yes, they really are unsafe after a week.

According to the USDA (similar information can be found from other food agencies):

Why do hard-cooked eggs spoil faster than fresh eggs?

When shell eggs are hard cooked, the protective coating is washed away, leaving bare the pores in the shell for bacteria to enter and contaminate it. Hard-cooked eggs should be refrigerated within 2 hours of cooking and used within a week.

Simply put, you're damaging the shell, and that expedites spoilage. Some secondary sources such as StillTasy (who use the agencies as their primary sources) are more liberal and say up to 2 weeks, but I'd at least give it a very thorough sniff test before eating hard-boiled eggs that old. (And note that the sniff test will not necessarily alert you to bacterial contamination, only the more conventional forms of food spoilage like rancidity).