All these buttons are just for formatting.  You can use the quote button when you want to quote something. The code button is useful for formatting code from an arbitrary programming language. The MCode button is useful for formatting Mathematica/Wolfram Language code. 

The Source button is a button that shows the post as raw markdown. It's useful sometimes when it is hard to use the buttons to format your post. 

How you take a screenshot depends on your operating system. You can often find tutorials online though with more information.

I normally choose to copy as Plain Text. If you have things which are heavily formatted in Mathematica, they may not be easily representable as plaintext which would make copying them hard. 

It sounds like your documentation center is not working. Are you able to use it all to look up information on functions? What operating system are you on?