Translation:I know that I am going to do well on my test tomorrow.
I wrote "in the test" and it corrected me to "on my test". I know that "in" is fine, but is "the test" also OK?
"In the test" is used in UK English or "at the test" I guess on the test is American english.
One of the options in the multiple choice, drop-down menu format was "Я знаю, что завтра хорошо напишет контрольную." Isn't that also a meaningful sentence (I know that he will do well on his exam tomorrow)?
Not really, not in this sentence: since the subject of the main clause is "я", the default assumption is that the dropped subject of the subordinate clause remains the same. So if you want to talk about him, it's either "Я знаю, что он завтра хорошо напишет контрольную" or "Он знает, что завтра хорошо напишет контрольную".
Even if the previous sentence was: "Он много учился."? I can't get a feel for when you can assume and drop pronouns or must specify.
Even in this case dropping "он" in a new sentence that already contained another subject would sound bad. You could, however, drop it in this sentence: "Он много учился и, я думаю, завтра хорошо напишет контрольную." - "He studied a lot and, I think, will do well on his test tomorrow." Note that in tis case you can simply drop "я думаю" from the sentence and it will still be a complete, normal sounding sentence. Hence "я думаю" does not really set or reset the default subject here.
Is the English translation "I know that tomorrow I will write good control."?
What does this mean?
It is not a control, but a test. And although the literal translation is writing a test, it is about taking a test (and knowing that one will do well on the test)
Is "at the test" instead of "on the test" fundamentally wrong or could it be said?
Interpreting this as a question about English, not the tranlation: I think you could say "at the test" but it would be less common, only in strict parallel with somewhat fixed phrases using "at".
Without listing endless fixed-phrases that use "on" or "at" or "in" but that don't really have any pattern of reasoning behind the choice of preposition, I can say that one pattern is that generally "on the test" will suggest focus on your test result, your grade. "At the test" might suggest 'during the test time, in the location of the testing place'. Ultimately they mean the same thing. But you could imagine the example of a first-time test proctor being asked if he did "well at the test". You couldn't ask the proctor if he did "well on the test". Unless he also took the test after/before proctoring it.