[KyleGoetz]

I'm only commenting on a few of the 取る／受ける question here.

Originally Posted by Maxful

一か月の休暇を取る
To take a month vacation.
(一か月の休憩を受ける)

The second sounds more like someone offered you a vacation, I think.

ケーキの最後の一切れを取る
To take the last slice of cake.
(ケーキの最後の一切れを受ける)

The second is like someone held a piece of cake out to you and you received it from him.

インターネットに試験を受ける
To take an exam on the internet.
(インターネットに試験を取る)

Where did you get this sample sentence from? I would have said インターネットで instead, and に feels off to me. But I'm assuming you got these from a list of native sentences, so I'm probably wrong.

お金のために仕事を受ける
To take up a job for the sake of money.
(お金のために仕事を取る)

I'm not exactly sure, but the first sounds like you're getting a job like we normally mean "to get a job" in English, while the latter is more like you're out networking and pulling in business yourself. Like one involves a boss saying "would you like to work here?" while the latter is you putting a ton of effort into finding potential clients. But I could very well be wrong here.

ペナルテイを受ける
To take a penalty.
(ペナルテイを取る)

I don't think the second one is correct. Furthermore, we don't say "take a penalty" in English. At least not in the US. We say "get" or "receive" or "commit." I'm guessing by "take a penalty" you mean when a referee comes up and gives you a yellow card or throws a flag or throws you out of the game. In that case, I think we say "receive a penalty" most often.