Translation:You cannot look at the textbook during the exam.
I'd argue cannot is more correct and what they're more likely to say - as in it's not even a possibility, you won't have them during the exam etc. Must not feels more like it's acknowledging that's an option, but you shouldn't do it, so it feels less assertive than outright saying "you won't be doing this!"
But yeah, either should be accepted here
I look at で as "in/during" and は as "as for", so transliterated it becomes "test" "in/during" "as for" -> "as for (in/during test)". This means that は encapsulates and makes テストで the subject ("in/during the test" is the subject).
Ultimately, its "As for (your time) during (the) test, ...". The は isn't necessary to my knowledge, since テストで also means "during the test". The は just draws extra emphasis that what follows deals specifically to "during the test".
To add to the other reply, Tae Kim puts this use of the は particle in perspective pretty well. He describes it as a topic (rather than subject) indicator, and translates it roughly like "as for [topic]".
In this context, it's there to indicate that you not being allowed to look at your textbook is specifically the case when you are taking the test ("as for [in/during a test], you cannot look at your textbook").
What is up with the seeming double use of topic marker in the Japanese sentence? Am i wrong to read テストでは as pairing で with テスト, and then pairing テストで with は again, so that the topic indicated by the は is "on the test"? Additionally, how would you break down the parts of 見てはいけません? So far, i figured that いけません was used as a verb inflection/conjugation, and not as a separate word by itself, like the separation of 見て and いけません here would indicate. I'm also unsure about how to think of 見て when it is modified by は. Would somebody please break this down for me?
Both the singular and the plural are correct. I'm a Mandarin native speaker, we usually don't distinguish between singular and plural either. (The concept of plurality is not that important in the languages' logic.) If necessary, we use words like "most" books, "part of the" books, or directly tell the exact number like "one (with a classifier)" book.
"You must not look at the textbook during the exam." I think this is better. ～してはいけません is stronger than ～してはなりません, like must and may/can (which tends to the Commonwealth vs. USA divide in my experience, though even I don't mind can in first-person, here it's just plain rude).