NoVEMber ve ocTOber hem khodaSHIM ba shaNA
I wrote, נובמבר ואוקטובר הם חודשים של השנה, but it said the answer was בשנה. Why "ב"?
Both sounds right to me, you can try report it. (but both are sentences I never heard anyone saying)
It now works (Dec 2019) to use the של construction
It didn't work for me, but I also had two typos in אוקטובר, even though it didn't point it out. Edit: Second time around and I see that you are correct.
Can we use it without הם?
No. In <noun(s)> is/are <noun(s)> type of sentences, you generally have to add a copula (הוא/היא/הם/הן).
It's frustrating to me that they ask for a translation of "of the year", but then their preferred answer is "in the year"