"It is cold. We want to eat hot pot."
I wrote the same sentence, but only left out 很. Shouldn't 天气冷 mean "it is cold" and 天气很冷 "it is very cold." ?
Although many people would immediately oppose, as a native speaker I agree with you that 很 is not mandatory in this sentence. Duo should accept it as correct. When the context provides enough information for the listener to conceive the magnitude of the adjective (it is as cold as one would think about eating hotpot), doing without 很 would not appear weird. On the other hand, for the same reason, with or without 很 does not really convey a different meaning.
In the tips and notes it was mentioned that for this course they would only accept 很（adjective）even though it may be correct without. It is correct without here, as whether with or without, it means about the same thing, really.
很 doesn't always mean "very". It's there for grammatical reasons in this case, linking the noun with the adjective. It works just like 是 except 是 is used for linking nouns with nouns. So in other words, 很 basically mean "is" in this sentence.
As a native I've always used 很 as very and in this case I would not use it, since there is no 'very' in the sentence. I honestly have never even heard of the two-syllable rule, I've definitely heard just 天气冷 around and about.
冷 is an adjective and as such follows a two syllable rule, so it is cold and it is very cold are both 天气很冷
I wrote 很冷 without 今天 on the front because the text in English was just "it's cold". Is it wrong?
It's too vague, I've never heard that being said on it's own. Something I have heard being said, though, which doesn't require 天气 or 今天 would be "好冷啊！＂which literally means" Wow it's cold!" which makes sense as an exclamation.
This should be translated as, thy weather is very cold rather than it is very cold if they're going to have us put 天气
I'm pretty sure 天气 is needed in the Chinese sentence, whereas "the weather" as the anticedent of "it" is often implied/understood in English.
It is the same as "We want to eat chicken nuggets" or "turkey and stuffing" or "pigs in blankets".