If an orthodox male Rabbi bestows semicha on a woman (dubbed Maharat), what is the status of her halachic decisions? This question is irrespective of whether or not the male Rabbi should confer semicha and whether or not the woman should publicly render halachic decisions. I am asking if, after the fact, Person X got a psak from a Maharat is that psak binding on Person X?
The issue we have to ask is why is a psak (legal ruling) ever binding in the first place? Why can't you just ask the next person?
There are two main possibilities (see Shach YD 242:31):
There a couple possible ways of testing this difference.
The Bavli (Avoda Zara 7a) tells that one may not ask a second authority (חכם) for a lenient ruling after receiving a stringent ruling from a first authority. The Yerushalmi (Shabbat 19:1 as understood by Rosh AZ 1:3) seems to say that one may not even ask a second authority for a stringent ruling after the first authority rules leniently. (Note: the Bavli doesn't explicitly reject the Yerushalmi's case; it just doesn't discuss it.) The Shach (at least) presents this argument in light of the two sides discussed above. Rama rules like Rosh's understanding of the Yerushalmi, while Shach disputes this, marshaling many, many authorities who ruled like the straightforward understanding of the Bavli.
If we take the first possibility, שויה אנפשיה חתיכה דאיסורא, then it seems there is little room to distinguish between who gives the psak, as the asker has accepted the answer upon themselves (provided of course there is no clear mistake made). If we take the second possibility, then we can distinguish between different authorities, depending on how much respect is due to a Maharat or other non-standard rabbinic authority. CYLO...A.
While I'm here it is worth noting that a women is certainly capable of issuing a "psak" if she is learned enough as is evident from the prohibition on issuing a psak after drinking wine. Sefer Hachinuch writes (#152):
When and to whom she should issue such rulings is, as you mentioned, not a part of this question.
Basically we have a rule, once chacham ("wise person") #1 has ruled specifically on your question, you're not supposed to ask chacham #2 unless you tell them your complete story. ("I asked chacham #1 and was told this, now how do you feel?")
My understanding is that this concerns actual ruling which requires making judgment calls on the case. If I call the Star-K and ask them "what bracha do you make on hearts of palm?" and they immediately reply "our policy is ha'etz", that wasn't a tailored ruling; if you really want to call the cRc and ask them, it's a bit silly but I don't think it's disrespectful.
As DoubleAA pointed out based on Sefer HaChinuch (and many great contemporary rabbis have ruled similarly), in theory a learned woman can render psak the same way a man can.
There are two remaining questions, really. Competence to render tailored psak, and communal structure.
The competence to render tailored psak simply depends on the knowledge and judgment of the person rendering it. You can't make a blanket rule about men or women with any given piece of paper. But please be aware that those who have spent much more time starting with the Talmudic theory and developing all the way through the sources can give a much better tailored psak than someone who was given some guidelines as a halachic digest. I don't know to what extent a Maharat is trained/authorized/expected to render tailored psak, and I suspect they'd be honest if you asked them this. Most yoatzot halacha, for example (including all those outside of Israel, as last I hear it), make no claims to issue binding psak, though they can consult a posek if needed. For instance, she could look at a bedika cloth and either say "this is obviously fine" or "this needs a judgment call, you need to speak to someone who can render psak." (A small handful in Israel have actually been given the advanced training to do so.)
Communal structure is a different issue; if my neighbor calls me up with a question that requires a judgment call, if there is a town rabbi who is available and competent to pasken then it's a good idea to direct the question to them. Ask me what happens to "communal structure" in a world where everyone just checks their own website instead, though; I don't know.
There is an issue of serarah brought by rambam in hilchot melachim 1:5 that women cannot serve in positions where they would have authority or direct command over others. L'chorah, issuing a psak Halacha is a form of serarah, so even if a woman is learned and became a Maharat, she couldn't, according to this De'ah.
Edit: I didn't realize this was such a charged subject- or was my halachic analysis really that distorted? There is an idea called serarah, it is brought down in the rambam, and it would preclude a woman from issuing a psak as binding, which is what the questioner asked. Of course, there are those that disagree with rambam. I do find myself more sympathetic to the idea of a Maharat than not, but issuing psakim is a complicated idea, when serarah is involved.