Removing one’s hair and nails:
The general rules:
Cutting one’s hair or nails with a vessel: If one cuts his hair or nails with a vessel he is liable. By hair he is liable even if he only cuts two hairs. If he cut only one hair he is exempt [from a Sin offering] although doing so is Biblically forbidden as is the law regarding all half measurements [i.e. that if one transgresses less than the measurement of liability he has nevertheless transgressed a Biblical command].
One who does not need to use the hair or nails: One who shears [hair] and cuts nails with a vessel is only liable [for a sin offering] if he needs the wool or the hair or the nails which he cut. However if he does not need them then this is a Melacha which is not needed for its own use [“Eino Tzarich Legufa”, which is only Rabbinically prohibited according to some opinions].
However there are opinions which say that even if one does not need the wool, hair and nails he is liable [for a sin offering] (as the principal Melacha [of shearing] is removing the hair and wool from the skin and the nails from the fingers, and he does need this removal in it of itself, and furthermore) also in the Tabernacle they would shear the skins of the Techashim even though they did not need their hair.
Removing ones hair/nails with fingers/teeth: One who removes his hair and nails with his hands or teeth is exempt [from liability]. However it is Rabbinically forbidden to remove [hair or nails] with ones hand even from another person, despite the fact that one is unable to train his hands to skillfully remove [the hair/nail] with his hands from another person, without use of a vessel.
One who removes a white hair from amongst black: One who gathers white [hairs] from amongst black hairs so he not appear old, even if he removes one hair, is liable [for a sin offering]. Doing so is even forbidden during the week due to the prohibition against “a man wearing the garment of a woman” which is defined as a man may not adorn himself with the adornments of women.
[See Q&A regarding cleaning out dirt from under ones nails]
Summary-cutting hair and nails on Shabbos:
Cutting one’s hair or nail with a vessel: It is Biblically forbidden to cut with a vessel even one hair [from anywhere on one’s body] or nail. However if one is cutting not in order to use the cut item, then there is a dispute as to whether this is Biblically or Rabbinically forbidden.
Removing ones hair/nails with fingers/teeth: It is Rabbinically forbidden to remove with ones finger or teeth even one hair [from anywhere on one’s body]. It is Rabbinically forbidden to peel or bite off a nail.
One who removes a white hair from amongst black: Is liable for a sin offering and doing so is forbidden even during the week due to the prohibition of “Beged Isha”.
May one scrape dirt from under his nail on Shabbos?
Yes, although when doing so one must beware to press down the flesh under the nail in order to avoid scraping the nail in the process of removing the dirt.
May one scrape off the inside of his nail?
No, as explained above.
May one pluck hairs out from a garment made of animal skin?
No. Doing so involves the Shearing prohibition.
May one ask a gentile to cut ones nails for Mikveh if they forgot to do so before Shabbos?
First opinion-Gentile may cut even with vessel: One who shears [hair] and cuts nails with a vessel is only liable [for a sin offering] if he needs the wool or the hair or the nails which he cut. However if he does not need them for this purpose then this is a Melacha which is not needed for its own use [which is only Rabbinically prohibited according to some opinions].
Therefore a woman which forgot to remove her nails before Shabbos and Shabbos is her night of immersion there is an opinion which permits her to tell a gentile woman to cut the nails for her, as all matters which are only Rabbinically forbidden are permitted to be done through a gentile for the need of a Mitzvah as is written in chapter 307 [Halacha 12] and 325 [Halacha 5].
Second Opinion-Gentile may not cut with vessel but may cut with teeth hands: However there are opinions which say that even if one does not need the wool, hair and nails he is liable [for a sin offering] (as the principal Melacha [of shearing] is removing the hair and wool from the skin and the nails from the fingers, and he does need this removal in it of itself, and furthermore) also in the Tabernacle they would shear the skins of the Techashim even though they did not need their hair.
Therefore one may not allow to ask a gentile woman to cut the nails being that this is a complete Rabbinical prohibition which was not permitted [even] for the need of a Mitzvah as was explained in chapter 276 [Halacha 8]. Rather one may tell the gentile woman to remove the nails with her hands or with her teeth which is a Rabbinical prohibition upon a Rabbinical prohibition. [See Q&A for the final ruling]
The reason why the above is not prohibited due to that she is assisting the gentile to cut the nails: Now, although she tilts her fingers towards the gentile and [thus] slightly assists her, this is [Halachicly] meaningless as one who assists does not carry [Halachic] weight (and is completely permitted Biblically. Even Rabbinically there is no prohibition against assisting, with exception to one who is doing a complete Melacha or a matter which is forbidden due to a Rabbinical prohibition not in the place of a Mitzvah. However in the case of a Mitzvah there was no decree made on the assistance just like they did not decree against telling the gentile [to do the Rabbinical prohibition] as he [the Jew] is not doing anything, and as well by assisting he is not doing anything as assistance truly carries no [Halachic] weight and it is just that the Sages decreed a decree against doing so due to that one may come to do so himself. However one who does an [actual] action is forbidden even if it is a Rabbinical prohibition upon a Rabbinical prohibition in a Mitzvah scenario, as was explained in chapter 331 [Halacha 8].)
Summary- May one ask a gentile to cut her nails for Mikveh if she forgot to do so before Shabbos?
A woman who needs to immerse on Friday night and forgot to cut her nails before Shabbos may ask a Gentile to cut her nails using her teeth or hands. Regarding if the gentile may use a vessel to cut them, this matter is disputed in Admur, see Q&A below regarding the final ruling.
Practically may one ask a gentile to cut the nails using a vessel or not? 
Rav Farkash rules that although Admur does not explicitly write to be stringent, nevertheless based on the rules of the Poskim, it is implied that he holds of the second, stringent, opinion, and one is to thus be stringent to forbid asking a gentile to cut the nails using a vessel even if she cannot do so otherwise. [For ruling of Mishneh Berurah and others see footnote]
What is the woman to do if a gentile is unable to remove the nails with her teeth or hands?
She is to clean out very well all dirt from under her nail and verify that there is no dirt left there. Care must be taken that she not scrape off part of the nail in the process. Thus she should press her finger downwards to open more space for the cleaning and thus prevent scraping.
May one ask a gentile to also remove the woman’s toe nails?
No. Rather she is to clean out from under her nail very well and verify that there is no dirt left there. Care must be taken that she not scrape off part of the nail in the process. Thus she should press her finger downwards to open more space up for the cleaning and thus prevent scraping.
This concept is called a “Melacha Sheiyno Tzarich Legufa”
 Ketzos Hashulchan 143/1
 Ketzos Hashulchan 143/1
 Admur 161/3, so rules also Biur Halacha there
 As scraping the nail is problematic due to the shearing prohibition.
 Ketzos Hashulchan 143/4 in name of Taz in end of 337
 Shach Nekudos Hakesef Yorah Deah 198
 Taz Yorah Deah 198/ 23
 Tahara Kehalacha chapter 19/79-3 footnote 202
 Vetzaruch Iyun as to why Admur did not rule this way explicitly as he does by other disputes?
 The Magen Avraham 340/1 rules that initially one is to have a gentile do so without using a vessel. However if this is not possible then the gentile may use a vessel to do so.
The Mishneh Berurah 340/3 rules like the Magen Avraham, although adding that the woman is to not assist at all in the removal of the nails, and the gentile is to be taught to take the woman’s hand herself.
 Taharah Kehalacha 19/79-3
 Taharah Kehalacha ibid in name of many Poskim
 Admur 161/3
 Taharah Kehalacha ibid in name of Mishneh Berurah in Biur Halacha
 As people are less particular regarding the cleanliness of the toe nails.
 Admur 161/3