Did the Aramaic of Jesus' milieu have 3 genders? If so, did the genders correspond to reality. I mean, were all inanimate objects neuter?

No, only two genders in Aramaic.

But note that even in languages with three genders it is very often the case that objects without gender are described by a noun with either feminine or masculine gender...

just as is the case in English, often objects without gender are described using pronouns with feminine or masculine gender.

Classic example: ships being by linguistic convention and tradition referred to using feminine pronouns.

This a side note to the answer.

1. Grammatical gender applies to the word -- the actual string of letters or sounds. Most languages that have it do not require that gr. gender match physical sex, even when there is a neutral gender. Often, though, gr. gender of words referring to people do match physical sex (but not always).

2. English has NO gr. gender (despite some people thinking otherwise). It does have words that refer to physical sex in the real world. Sometimes, they are used figuratively, like using she/her for boats, BUT there is a huge difference from gr. gender.

In English, the word chosen doesn't matter. I can use ship, boat, craft, vessel, yacht etc, and still use she/her. That's a figurative use of physical sex.

In a language with gr. gender, there is no guaranteed that different words for the same object will have the same gr. gender. Gender is independent of meaning or the real world.

It takes a while for a native English speaker to grasp the difference. he/she/it -- are not examples of grammatical gender. Neither are boy/girl, bull/cow; actor/actress. They all refer to physical sex.

In French, my name, since I'm a guy, is assigned a masculine gender to match my sex. However, if I use a different noun, like man or person, then the gr. gender of those words is used to describe me, until a new noun (perhaps my name) is used. man - is masculine, but person is feminine. Using person to describe me in no way suggests I am female. They are separate concepts.

Koine Greek also had grammatical gender.

Note: Languages with gr. gender can also have words for physical sex. Example from French:

chatte - female cat (feminine gender). chat - male OR female cat (masculine gender).

vache - cow (female bovine; fem. gender). taureau - bull (male bovine; masc. gender)

Like in English, there are only two genders, male and female. But also like in English there is a "genderless" pronoun. In English that is "it". Something referred to as "it" is not assisted a gender.

And like in English, object that are actually genderless can be tradition be referred as a male or female. Such as ships being referred to as female, or a person whose gender is unknown being referred as male (King's English.)

If you are looking to try to argue that Jesus allowed for more genders then male or female in his teachings, you will not find any such thing in Aramaic. But even if there was, all of the sayings of Jesus are recorded for us in Greek and not Aramaic, so we have no way of knowing what Aramaic pronoun Jesus might have used for something. Only the male or female pronouns of Greek.