"After he heard the news, he cried sadly."
doesn't 这个 mean 'this'? Shouldn't it be, "after hearing this news"? Or are there instances where 这个 means 'the'?
Exactly. We know 这个 can be "this". But if it can also be "the" then it follows that when translating from the English given we should be able to use either "the" or "this" and when translating from the Chinese given we should be able to use or leave out 这个.
It is barely alright, because you have left out 這個, the clauses still maintain a close correlation. If you add back 這個 and some other details, the sentence will begin to fall apart. You would need some "glue" to relate the clauses tightly e.g. 他听了这个令人震惊的新闻之后，难过地哭了。
The sentence syntax in Chinese is like a tree. You draw the stem first and then the branches, and then the leaves. Here the "stem" is 他哭了. 听了这个新闻 is a "branch", describing in which circumstance the stem meaning happens. It is why the 他 in this part is omitted rather than the latter one.
他聽完了新聞以後，她難過地哭 should also be acceptable too... you can leave out 這個 in this case too.
·shouldn't the correct answer include 以后 to indicate "After"？like this 他听了新闻后难过地哭了
了 indicates the state that the news was heard. Arguably we could translate the sentence otherwise
e.g. Having heard that news, he... So without 后 or 以后 it can still be correct.