STEM undergraduate non-STEM master, advanced degree pool Eligible during OPT extension?

Hi, I finished a BA in Computer Science from a Canadian institution and will go directly to the US for a non-STEM master degree. Just to make sure after graduation I will be eligible for STEM extension? And if does my employer need prove my work is related to my STEM degree?

And also, during the STEM opt extension period, can I get lottery in the advanced degree pool during both the second/third lottery?