As one of the other answers stated, 2333 = hahaha. Also, xswl (short for 笑死我了) = laugh to death

Speaking from experience, as long as you don't focus too much on the Pinyin (as in only reading the Pinyin) there shouldn't be any harm. The Pinyin is only supposed to familiarise you to the words, ...

I think yours is not wrong, but usually I see '蓝天白云 晴空万里' being used to describe clear, beautiful weather(clear blue sky with white clouds). Also '晴空万里无云' (clear blue sky without clouds)

It means 'able people usually get more work and therefore more busy/tired'. The English equivalent would be 'With great power comes great responsibility.' Usually used as a compliment like this: A: ...

It's 「天佑美國」. As for 佑 vs 祐, take a look here