"They are people and we are people!"
Translation:One są ludźmi i my jesteśmy ludźmi!
Usally - not really. You rather omit the pronoun because in most contexts it's totally redundant, so you just choose it to emphasize or to contrast it with something else, mostly when changing the subject during the sentence.
Here, maybe there's no contrast, but there's the change of the subject. So without "my", it would feel quite strange.
And, indeed, both are accepted. The thing is, though, that in the event the some other error is made, the first correct answer in Duo's list of correct answers is the "One są..." version, and so that is the one which will be shown to the learner. In so doing, Duo is not telling the learner that "Oni są..." specifically is wrong; just that something is wrong somewhere.
"To" isn't really used with personal pronouns. The verb "być" is better when referring to people. "To" may be used for animals, but it is often used with inanimate objects.
Ex: Ten budynek to apteka: This building is a pharmacy
Tygrys to kot/Tygrys jest kotem: A tiger is a cat
Yes, you are wrong. The first pronoun "one" or "oni" cannot usually be dropped, because the third person (both singular and plural) often indicates the gender and can only be dropped when previously introduced and absolutely clear (there is no such context here). It cannot be dropped also because the sentence is contrasting the grammatical subjects "oni" and "my" and in this case, both have to be explicitly included, not only the second one.