As far as I can tell, this sentence can mean both "there are a number of different places in which we go to eat", and "me and someone else - we dine in separate locations"; is there a way to discern between the two?
I would assume the placement of "diversi" changes the meaning, much like other words. For example, "un grande amico" means a great friend, whereas "un amico grande" is a large friend. Also, there is the option to use "differenti", which may change the meaning as well.
That seems like some sound logic! My guess is that, just like the English translation "We eat in different places," it would all depend on context and intonation. :)
Mmm...that sounds awkward, unless you are talking a set amount of specific places, but, in that situation, I think «Mangiamo in quei diversi posti.» would sound better.
Very few adjectives can come before the nouns. «diverso» is one of them, I guess. I do not believe it would be incorrect to say «Mangiamo in posti diversi.».
Perhaps because "diverse" sounds strange in that usage. You get the same meaning and a less strange effect by using "various" or "several."
Here, the speaker is not talking about specific places; they are speaking in general. This is why «nei» («in» + definite article «i») is not used; the definite article is only used when talking about specifics. The speaker is not saying "We eat in the various places [previously mentioned];" (s)he is saying "We eat in various places."
Would it mean the same if I said "Mangiamo in parecchi posti"? Or "Mangiamo in molti posti"?
No. Different stuff.
Parecchi means 'various/different'; Similar to 'altro/altri'.
Molti means 'many'.
Duolingo's translation for "diversi" is "various" in this exercise, so it does look like it means the same as "parecchi".