I guess Duolingo's hints misdirected you. Here are a few more: http://pt.bab.la/dicionario/portugues-ingles/tratar. Try looking it up in this dictionary: http://michaelis.uol.com.br/moderno/ingles/.
Here are my thoughts for the little they are worth. Although "tratar" is translated here as "talk about", a closer translation is "discuss" (which sounds like a slightly more serious activity). So we can discuss (use "tratar"), talk about (use "falar") or say something about (use "dizer") a subject.
'To treat with' used to be a formal phrasal verb, as in, negotiate a treaty: 'We Spanish treated with the French'. Most English speakers have never in their lives used this verb phrase, written or spoken. But it's where one modern version comes from: I treat you well, treat them with respect etc. ;-)
Yes, although many common idioms use "treat with" (”treat with kid gloves" comes to mind) and it seems more familiar than "treat of" I don't think I"ve ever used it in the sense of this dictionary entry: http://www.macmillandictionary.com/dictionary/british/treat-with.
I don't think it's ever used in the first person, because the meaning is closer to "is about" or "is concerned with." I gather the Portuguese sentence means "I'd like to discuss the calendar" of events or work that the speaker and his or her colleagues have planned. But without that context it's very hard to figure out what it could mean. In that sense we might just as well say "I want to deal with the calendar," meaning discuss the question of the calendar.
I couldn't remember coming across "treat of" before you mentioned it and I tried to find a suitable dictionary definition. It is missing from several online dictionaries but from what I found the meaning really is closer to "deal with" than "is about". See, for example: http://www.collinsdictionary.com/dictionary/english/treat (verb - item 8).
In fact, it does appear possible to use it in the first person. Herman Melville writes in Moby Dick: "But this august dignity I treat of, is not the dignity of kings and robes, but that abounding dignity which has no robed investiture". So perhaps you have hit upon a possible (very formal) translation after all.
Good work, Davu. Thank you for hunting down that citation from Melville. You see, however, that using that phrase is very literary and archaic, especially in the first person, and we would today more often say "deal with" instead. That also gets close to the basic meaning of "tratar" as used in other contexts. Both phrases are loosely synonymous with "talk about," so they all convey the sense. I would plump for "deal with" here as the best rendering, assuming that the context imagined for the speaker is some kind of issue to resolve.