What is the ruling on throwing nail clippings in the trash, and what is the ruling on burning hair that falls from a comb, and is what some women do, using and undoing a rosary in the mosque to hasten marriage, permissible in Sharia?

Scholars have recommended burying detached nails and hair. Imam Ahmad was asked about this and he commanded that they be buried. As for burning hair, there is no prohibition mentioned, and the basic principle is permissibility, but burial is preferable. What you mentioned about taking a rosary (subhah) and reciting Al-Fatihah or Al-Ikhlas seven times after cutting hair has not been reported in the books of Sunnah. This is an innovation in religion, and it is a forbidden Bid'ah (innovation). A Muslim should avoid such things and repent to Allah, and turn to Allah with supplication, for He (the Almighty) said: "And when My servants ask you concerning Me, then indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me and believe in Me, that they may be guided," and He said: "And your Lord says, 'Call upon Me; I will respond to you.'"