Long post today to make up for not posting every day.
Tosafos explains that we pasken like the opinion of Rav Huna that one is in violation for being makif a child, even though the child is not yet obligated in the prohibition. Although R' Ada understands that whenever the subject (nikaf) is not obligated, there is no prohibition on the makif either, R' Huna seems to reject this view. Tosafos explains that R' Huna rejects the drasha that exempts a makif whenever the nikaf is not obligated. I would like to suggest another explanation for the opinion of R' Huna. The categorization of a child not being a bar chiyuva (or not being a bar onshin), implies that a child is not considered in the realm of the mitzvah at all. Perhaps R' Huna understands that a child may not be obligated but is in the realm of the issur, therefore he is very different from a goy for example who is not at all shayach to this issur. Although R' Huna may in fact agree with the connection of R' Ada that whenever the nikaf is not a "bar chiyuva", the makif is exempt, he disagrees with the classification of the child not being a bar chiyuva in this respect.
Tosafos goes lishatasam. Based on their understanding that R' Huna rejects the hekesh of R' Ada making the makif dependent on the nikaf, asks why R' Huna would permit being makif a goy. Tosafos creates a new drasha to answer this, that "pe'as rosh'chem" in the pasuk is written only by jews, not by non-jews. Based on this approach Tosafos can't find a legitimate source to be matir a man being makif a woman, and therefore leaves it as a question whether it is in fact permitted for a man to be makif a woman. However, based on the approach that i suggested above that R' Huna agrees to the drasha, just that it doesn't apply to a child since the child is relevant to the issur albeit not "obligated". To be makif a goy or a woman would certainly be included in the hekesh, and R' Huna would agree that it is permitted.
In Y.d. 181:5 the shulchan aruch paskens like R' Huna that one cannot be makif a child. The Rama writes that one may be makif a goy or woman, but then cites those who are mesupak about that. The Gra quotes 4 opinions: 1. The Nemukei Yosef is mesupak whether one can even be makif a goy according to R' Huna. 2. Tosafos says a goy is pashut that it is mutar to be makif, and is mesupak about being makif a woman. 3. Tosafos in Shavuos seems to take for granted that only according to R' Ada would there be a heter to be makif a woman, but according to R' Huna it would be assur. 4. Rosh holds that one can be makif a goy and even a woman, even according to R' Huna, because the only issur is to be makif a kattan who is "אתי לכלל חיובא".
The Rosh seems to hold like the sevara that i suggested. R' Huna agrees that there is no issur on the makif whenever there is no issur on the nikaf. However, if the nikaf is a child there IS an issur on the nikaf even though he is not yet old enough to be muzhar. BUT if the nikaf is a goy or woman who are inherently excluded from the entire issur, there isn't any issur on the makif either. The Rama is matir based on the Rosh, and the Shach supports this opinion and comments that the bach and other ommit the opinion who is machmir about this.
Regarding a woman doing hakafa to a man, following R' Huna it should be mutar. The Shulchan Aruch paskens that it is mutar, but then says "yesh omrim" that it is assur. R' Akiva Eiger points out that even if we assume that there is no issur for a woman to be a makif, it would be assur for her to be makif a man since she is in violation of a d'rabonon of lifne iver (not d'oraysa since it is not trei avri d'nahara), but is then mesupak that since by her doing it, she will prevent him from doing it himself and violating a greater issur, she is actually saving him from an issur and therefore would even be mutar m'darabonon. He then suggests that even if it would be assur for a woman to be makif a gadol, it would be permitted for her to be makif a kattan as is apparent from the ma'aseh of R' Huna.
B'kitzur, if a father wishes to cut off his sons peios, he can't do it himself, nor can he tell a goy to do it because of amira l'nachri (pischei teshuva). The mother can do it herself, and can certainly tell a goy to do it . However, the proof that a mother can do it is from the story of R' Huna where Chova did it. Rashi explains that it was done as part of a refuah process. Therefore, if it is not being done for refuah, there isn't any proof from the gemara and it may in fact be assur m'drabonon. Also, if the hakafa to a child qualifies as hakafa, it should be assur even for a mother to do it because she is literally feeding him issur, just as she can't feed him any other issur. This is a question on how chova was able to do hakafa to her children, why do the meforshim (including R' akiva eiger) fail to mention this? I would like to suggest a big chiddush based on this question. The issur to be a nikaf although it doesn't require an action (except to be chayav malkus - shulchan aruch 181:4), it does require consent. The issur is not being in a state of hakafa where one's peios were cut off, rather the issur is to allow yourself to be shaved. Therefore, when one is makif a child, the child is merely a subject that hakafa was done on, but since the consent of a child doesn't qualify as consent, he is not considered a nikaf and shaving his head is not considered feeding him issur. In contrast, if one would bring a kohen child into a cemetery, it would qualify as feeding him issur as the poskim say since the issur is the effect of the tu'mah, not the decission to become tamei.