Is there anything prohibiting/restricting a woman from reading Pirkei Avot or the Shulchan Aruch?
Rav Moshe Feinstein zt"l in Igros Moshe, YD III, siman 87 (top left column) famously said that girls can learn pirkei avos:
משום שהוא עניני מוסר והנהגות טובות יש ללמדם בהסבר לעוררן לאהבת תורה ולמדות טובות
Because it inspires mussar (ethical self-improvement) and a good way to act. It must be taught in an explanation that inspires them to love Torah and gain good middos (character attributes).
As far as learning Shulchan Aruch, it is clear that women should learn the halachos that are relevant to them. This codified by the Rema in YD 246:6:
ומ"מ חייבת האשה ללמוד דינים השייכים לאשה
And nevertheless, a woman is obligated to learn the laws that are relevant to women.
There are different opinions when it comes to women learning Torah, however I believe everyone allows for women to learn Pirkie Avos. With regards to Shulchan Aruch, all laws that apply to women I believe everyone would agree is allowed as well.
I wrote up some sources a few months ago which I'll place below, however I don't recall if it's I complete copy and past from somewhere else or a combination from a few places so I'm sorry in advance if this is a rip of from someone else.
The Gemara (Kiddushin 29b) teaches that women are not required to study Torah. The Mishna (Sotah 20a) records a celebrated dispute between Ben Azzai and Rabbi Eliezer if it is advisable to teach women Torah. Ben Azzai believes that one should teach his daughter Torah and Rabbi Eliezer strongly advises against it. The Mishna concludes with Rabbi Yehoshua's statement which supports Rabbi Eliezer's view. The Rambam (Hilchot Talmud Torah 1:13) and the Shulchan Aruch (Yoreh Deah 246:6) rule in accordance with the opinion of Rabbi Eliezer and Rabbi Yehoshua.
Even though the Chafetz Chaim endorsed the Bait Yaakov movement he did not specifically approve the teaching of Gemara to women. He explains his position in his commentary to Sotah 20a (in his Sefer, Likutei Halachot): It seems to me that this [issue of refraining from teaching Torah to women] applies only at those times in history when everyone lived in the place of his ancestors and the Mesora was very strong for all, and this constituted sufficient motivation for everyone to act in the same manner as their forebears....In this type of situation we can say that women should not study Torah and women will learn the ways of Torah by emulating their righteous parents. Today, however, when the Mesora has become very weakened, and it is common for people to live far away from their parents, and women learn to read and write a secular language, it is a great Mitzvah to teach women Tanach and the Mussar of Chazal from Pirkei Avot and Menorat Hamaor, so that the truth of our Mesora will become evident to them. The alternative, God forbid, is for women to deviate entirely from the way of Hashem and Torah.
The Satmar Rebbe (Vayoel Moshe pages 451‑452) points out that the Chafetz Chaim limited his permission to the study of Tanach and Mussar. Similarly, Rav Moshe Feinstein (Igrot Moshe Y.D. 3:87) rules that women may not study Mishnayos with the sole exception of Pirkei Avot.
Those who believe women should study Gemara
Rav Yosef Dov Soloveitchik taught women Gemara at the inaugural Shiur delivered at the Stern College Beit Midrash and instituted girls' study of Gemara at the Maimonides School in Boston. Rav Meyer Twersky (Tradition Summer 1996 pages 98‑99) explains the Halachic basis for his grandfather's practice:
The Halacha prohibiting Torah study for women is not indiscriminate or all‑encompassing. There is complete unanimity that women are obligated to study Halachot pertaining to Mitzvot which are incumbent upon them (Rama Y.D.246:6). Clearly, men are permitted to provide instruction in these areas. A father's obligation of Chinuch relates equally to sons and daughters (see, for example, Yoma 82a). The prohibition of teaching the Oral Law to women relates to optional study. If ever circumstances dictate that study of the Oral Law is necessary to provide a firm foundation for faith, such study becomes obligatory and obviously lies beyond the pale of any prohibition. Undoubtedly, the Rav's prescription was more far‑reaching than that of the Chafetz Chaim and others. But the difference in magnitude should not obscure their fundamental agreement: intuitively, it is clear that the guidelines of the Talmud in Masechet Sotah were never intended for our epoch. This is not an instance of modernism, but Torah intuition.
Rav Eliezer Waldenberg's Teshuvot Tzitz Eliezer (9:3), based on the Chafetz Chaim appears to concur with that of Rav Twersky.
In September 1986 a number of students asked Rav Aharon Soloveitchik (after completing a Shiur delivered at Yeshiva University) his opinion about teaching Gemara to women. He responded by citing the Torah Temima (Devarim 11:19) who quotes the Teshuvot Maayan Ganim. This authority writes that Chazal only prohibited coercing women to study Torah. If, however, they choose to learn Torah, then they deserve full support of the community.
Rav Chaim David Halevi (Teshuvot Asei Lecha Rav 2:52) adopts a similar approach. He writes that high school girls who wish to study Torah are permitted to do so. He cites the Chida as supporting this view (also see Prisha to Y.D. 246). Rav Yehuda Henkin (Teshuvot Bnei Banim 3:12) points out that this approach explains the phenomenon of women in various generations who were Torah scholars. The examples of outstanding women Torah scholars include Bruriah (see Pesachim 62b and Eruvin 53b), the mother of the Prisha (whom the Prisha occasionally quotes and notes that she was proficient in many areas of Halacha), the grandmother of the Maharshal, the wife of the Netziv, the wife of Rav Isser Zalman Meltzer (regarding whom it is said that she helped edit the Even Haezel ‑ Rav Isser Zalman's commentary to the Rambam) and Nechama Leibowitz.
Rav Aharon Lichtenstein often points out that Talmud Torah promotes Ahavat Hashem (see the Rambam's Sefer HaMitzvot, Mitzva number three). Thus he argues that women who study Torah have the opportunity to fulfill the Mitzva of Ahavat Hashem. Rav Lichtenstein and Rav Henkin argue that women who study secular studies at the highest level should also study Torah at the highest levels including the study of Gemara.
Historically, there were many wome who studied Torah. Currently, we have women who teach Torah, and Rabbi Avi Weiss has given some women the title "Rabbanit". Women are allowed to be mashgichot (kosher supervisors), so, I assume that they are learning some parts of Yoreh De'ah to know what to do.
I am citing these as examples of what appears to be a permissibility. However, there are so many differing opinions on this subject, that are all credible. You may want to read this article which throughly addresses the diferent opinions on this issue. I decided not to excerpt anything from it, as I think all the opinions mentioned are valid, and I can't decide which ones I like best.