PAA,
Sorry for the delayed response.
“So what role are they violating by answering halachic questions?”
I don’t think women are violating a role by answering halachic questions (assuming they are learned enough to answer correctly). However, chazal specify that talmud Torah is a mitzvah particular to men, and women are only required to learn the Torah that is relevant for their role as mothers and wives (and nowadays for some, by necessity only, as persons of business). Just from that detail alone it is objectively apparent that dedicated Torah scholarship is a role halachically and traditionally intended for men.
Additionally, regarding the original question of, ‘Why can women not become rabbis?’: Assuming you define rabbi in the colloquial sense of one who acts as a dedicated higher level gemara teacher or one who acts as a community pulpit rabbi – you run into tznius issues. I don’t think you will disagree with the claim that it is inappropriate for a young ordained female to take the pulpit in front of a congregation of men to deliver a weekly sermon, to lead a davening (even from the other side of the mechitza), or to teach gemara to a classroom full of hormonal high school boys for several hours daily. For any other position, a smicha is not required, and then the question begs: If a smicha is not required for any position that a woman may be permitted to take, then why is it being demanded with such emotion and force? I believe the obvious answer is that it is not a totally altruistic reason as some may have you believe.
Finally, it seems to me that an easier answer to your question is available: Our gedolim and lesser Torah leaders have implicitly or explicitly come out against those who try to give smicha to women. Presumably, these Torah scholars have more da’as Torah than most (read ‘all’) on this forum, and they hold it is against da’as Torah. If an un-ordained woman thinks she has more da’as Torah than those in positions to grant ordination unto others, there is clearly something amiss.
“According to the teaching that they will teach you and according to the judgment they will say to you, shall you do; you shall not deviate from the word they will tell you, right or left.” [Devarim 17:11]