For me, the most natural verb is "have" (tomorrow we will have a test), but I can see a small semantic difference. Should "tomorrow we will have a test" be accepted?
Long ago, when I was at school, it was always "we are going to have a test tomorrow" or "you will have an exam" etc - I am not sure, however, if this is British English, Scottish English or just a quirk on my island where the use of English is heavily influenced by Gaidhlig.
That would be typical of American English as well, along with "take," although the two have a slightly different feel to me. If I say "you have a test," it sounds to me as if I am simply talking about the appointment itself. I could also say "you have school," "you have a haircut," or "you have a dental appointment." On the other hand "you are taking a test" sounds much more active to me, as it is talking about the actual activity of taking, or as the English would say sitting, the test itself.
Even though that's the same meaning I feel like the translation is not accurate. Będziemy pisali should translate by "we will be writing", however the options let you choose between "we will be taking" or "we will write". " We will be taking" is accepted as a correct answer even though it's not the right verb. Am I missing something or is it a problem in the exercise?
This is an English-language thing: in America, we say the students "take" a test and the teacher "writes" it, but in Canada and much (all?) of Europe, the students "write" a test. The Polish sentence is refering to what the students do, but either verb is ok in English, depending on your region.
That's not accurate. In the UK we say the student "takes" a test (indicating an action), or "has" a test (if the act of taking the test is in the future). If we were talking about a teacher creating a test, we would probably use "creating" or "devising", but "writing" might be okay at a pinch. Then the act of giving the test to the pupils is called "setting a test for...". I translated the answer literally as "writing", but if the context is that of a pupil, then the best (British) translation would be "taking".
It's perfectly correct, and I might well say it myself. In British English "shall" and "will" are usually interchangeable in normal 1st person singular and plural sentences.
But as far as I know, this is not so common in American English, and even in Britain we're a dying breed. Fewer young people are using "shall" in this way nowadays, I believe.
I believe this is because this Polish sentence does not mean what the US English sentence "We will be writing a test" means. This is spoken by a student. In US English, the teacher "writes" the test and the students "take" it. I believe in UK English, the student might "sit" or "sit for" a test, but I'm pretty sure it is still the teacher who "writes" it. The student might "write" an essay on an exam or "write" a particular answer, but he "takes" the test as a whole.
I think that zdawać is rarely used with test, one usually says zdawać egzamin and pisać test. But nothing of a big difference.
Zdawać has, by the way, two meanings. One is to take/have/sit, and the second is to pass (the exam). But this is not ambiguous in practice because one rather uses zdać (perfective form) in the meaning of passing it.
You can always use the infinitive in such constructions, unless it results in having a double infinitive, which is rather rare. The infinitive doesn't imply any gender. "będziemy pisać" works here. A typo could make the algorithm correct you to the main answer, for which someone chose "będziemy pisali" (at least one man among us).
Or alternatively, you can use the gendered past tense verb. The infinitive is surely easier to use for a learner, but you need to be aware that the gendered version is also everyday usage.