Learn Farsi: می‌خواستند سفارت را شنود کنند، ولی ناموفق بودند. - They were going to wiretap the embassy, but they were unsuccessful.

Yes, literally it translates to "they wanted to wiretap". However, in Farsi, the verb خواستن (to want) in the past continuous tense (می‌خواستند) is frequently used to express a past intention or an action someone was just on the verge of doing. In English, this is most naturally translated as "they were going to" or "they were about to".

How does the compound verb شنود کردن form the meaning of "to wiretap"?

The noun شنود (shonud) means "eavesdropping" or "audio interception." It shares the same linguistic root as the basic verb شنیدن (shenidan, to hear). When combined with کردن (to do), it literally means "to do eavesdropping," which is the standard term for wiretapping or bugging a room.

Instead of using the negative adjective in ناموفق بودند (they were unsuccessful), could I use a negative verb like موفق نشدند (they did not succeed)?

Absolutely, both are perfectly natural and mean the same thing. Using the negative prefix نا- (nâ-) with the adjective موفق emphasizes their state ("they were unsuccessful"), while negating the compound verb موفق شدن emphasizes the outcome of the action ("they did not succeed").

Since this lesson covers double negation for strong affirmation, how would I change this sentence to say "They couldn't NOT wiretap the embassy" (meaning they had absolutely no choice)?

You would combine the negative past continuous of توانستن (to be able to) with the negative subjunctive of the main verb. The sentence would become: (ne-mi-tavânestand sefârat râ shonud na-konand). This double negative perfectly expresses that they were absolutely compelled to wiretap it.