They are a mono-gendered species that don't recognize the concept of separate gender roles. Femininity is not a universal behavioral attributes across different cultures or even species. In fact, a lot of Asari characters you interacted with are portrayed quite neutrally. You can replace a lot of them as males and their characterization didn't even suffer..... except for the nightclub dancers and prostitutes, I suppose.
Asari probably don't have a use for all the words you used to describe gender, except perhaps to describe weird alien mating rituals.
For them there's just Asari, everything else connected to the idea of more than one gender has to so with aliens or possibly animals native to their home-world.
As for dancers and prostitutes, I'm rather certain that there are males in these roles as well, so I don't see your point.
I won't argue the second part, the Asari have as much lore as any other main race in the Mass Effect universe, but as I see it, it doesn't change that in their very core they are in the game to serve as sexual objects.
What I really liked though is that with Javik Bioware threw hints that the Asari are like that because they were engineered to be like that by the Protheans. It's like an in-game explanation to why they are fan service. Of course, Javik just hints at Asari being engineered sex slaves and I don't see that as a fact, only as a theory. But it's one that fits.
"Sexual object" means that someone is there for the sole purpose of looking sexy, like when they use female or male models to sell a car, a deodorant, or anything else.
A fully fledged character that also looks sexually appealing (even if it is to the "wrong crowd"), has nothing to do with it.
Although admittedly sometimes they go overboard with some aspects, like with Miranda, but that's the Bio tendency of being subtle as a hammer to the face.
The Asari are there for much more than that, they are a fully fledged race, equal to the other races, and in a position of authority over the rest of the council. Ditto for Liara, her looks are far from the main thing about her character.
As for Prothean genetic vandalism, it is actually an older fan theory that was needed not only to explain the Asari, but also the inexplicable prevalence of so many humanoid races with some similarities to one another in the same galaxy.