Having long nails is against the fitrah. The reason behind cutting the nails is cleanliness and hygiene because dirt can gather under the nails.

Removing the pubic hair, whether by plucking, using a depilatory agent, shaving or cutting, is one of the actions dictated by the fitrah and encouraged by Islam, but it is not restricted to or required at the end of every period.

In Islam, there is nothing wrong with throwing cut nails in the garbage or burying them. But if one is afraid that they may fall into the hands of a practitioner of Sihr, then he should bury them or throw them somewhere where such people will not be able to get their hands on them.

Sunan al-Fitrah are five things which, if they are done, will be in accordance with the natural pattern on which Allah made mankind and urged them to follow, so that they will be better and more perfect. For more, see the detailed answer.