Is it obligatory to get water under fingernails that are attached to the skin, and under the chin, and is the Hadith "that the Messenger of Allah, peace and blessings be upon him, when he performed ablution, he would take a handful of water and put it under his chin and run it through his beard, saying: (Thus my Lord, the Almighty and Exalted, commanded me)" authentic, and how can one ensure that dirt does not prevent water from reaching [the skin] during ablution?

It is not necessary to use something to lift the nail to get water underneath it, and a small amount of dirt under the nail is excused. As for the area under the chin, it is not part of the face, so washing it is not obligatory. The Hadith about running fingers through the beard (takhleel al-lihya) is disputed, and even if it is authentic, it is for recommendation (istihbab). Materials that prevent water from reaching the skin are those that have a "physical body" (jirm) and form a thick layer. However, if a material does not have a physical body, or if it is a small physical body that is removed by water, it does not prevent the validity of ablution.