In your first sentence, 他打开车门，拿出一个书包后又把它关上了。 又 doesn't mean "again". So it's not worth comparison with "再". It means ”but", but a slight transition. When they both mean "again". 又 is the repetition of a "similar" action or situation while ”再" is the addition of an action or situation.

又表示 又一次做这件事；再表示一个次序，先做一件事，再做一件事。两个并不完全相同。

Both 又 and 再 are adverbs meaning "again." Generally, you place the adverb right before the entire verbal phrase in Chinese, so subject + adverb + verb + object. Think of "把它关上了" as the entire verbal phrase. You add 又 or 再 before the phrase, to get "又／再把它关上了." However, for whatever reason, in the second sentence you cited, 再 comes after 会。 In other words, while it is 又会不高兴了, you would definitely say 会再不高兴了.

再 means again / repeat 又 has two meanings: 1) 再 2) highlight the emotion of speaking. for example, 你失败了一次又一次。 he lost again and again. 他昨天迟到了，今天又迟到了。 he was late yesterday, and he was late again today. in ur two phrases, 又 is used as its second meaning. 要是他知道这件事，会不高兴了。-------without 又, u are just telling us a fact. 要是他知道这件事，又会不高兴了。------with 又, u are telling us this fact and highlighting that he gonna be unhappy.

You can look up 又 再 in 现代汉语词典（第6版）Now I must to go to school for evening class.^_^