My Chinese name is pronounced a lot like the word "June." So because of this I chose "June" for my English name.

It flows a lot better how it is. Either works in common conversation. I'm not sure how grammatically correct the 2nd option is, though.

They aren't anything you NEED to know, but they're nice to know. They all come from things that are cultural. "Best thing since sliced bread" means that something is a really great and convenient thing. It means this because the invention of slicing bread was super convenient. "Cost an arm in a leg" is an exaggeration- meaning something is so expensive or hard that enable to get it you would have to give up an arm and a leg (but not literally.) etc etc

it is not correct, because of the "ing" on "working." if you wish to use "working" you need to specify it... like "During my time working at Nanjing University..." or "At the time I was working at Nanjing University I was..."

안녕하세요, 제 이름은 name goes here 입니다. Annyeonghaseyo, je ireumeun name goes here imnida.