"He became president."

Why is there no article before 'president' ??
And can other cases go the same way? (such as teacher, golfer, doctor, soccer player...)

See a translation

There is not an article because the writer might have done it for a reason. I'll explain.

"He became president" or "he became A president" or "he became THE president", they are all correct depending what the reader already knows and what the intended message is of the writer.

After his long career in politics, he eventually became president. This is assuming that the reader knows who you are talking about.
He became A president of the fan club. When written this way, it tells the reader that there are other presidents as well.
He became THE president. This assumes the reader knows what he is now presiding over as president. If it is not clear what he became the president of, then you would want to share that information.
He became the president of the company.

Yes you can apply the same explanation to the others you mentioned.