1. The shapes used to be on the shape menu! They were moved to the ruler! So the difference is, on the ruler the shapes are basically a selection which you can use to trace around.
2. Good question, why don't we have 'basic' shapes we can fill in the Shape menu. My guess is that when Adobe moved the tool they didn't think we'd still want it as something we could easily fill. So here's what you do with the selection tools in the ruler. Create your shape, change to the fill tool, make sure you have the colour you want then just fill the shape, you'll get a Vector/Pixel popup choice.
3. I'm not really sure what you mean by select and save a colour. The used colours should all appear in the recent area. Where are you saving colours from/to?