Praise be to Allah.
You do not have to use something to lift the nail so that the water can pass beneath it; rather that is a kind of strictness that is not acceptable according to Islamic teachings. If the person who is doing wudoo’ washes the parts of his body in an ordinary manner, then he has obeyed the command of Allah, may He be exalted, and has done what is required of him.
If it so happens that there is a small amount of anything under the nail that the water does not reach, then it comes under the heading of negligible matters that are pardoned, such as dirt under the nails and cracks in the feet; the one who is doing wudoo’ does not have to make sure that water enters every single crack or to check them and examine them.
It says in Mataalib Ooli an-Nuha (1/116): A small amount of dirt under the nails and the like does not matter, and the same applies to the inside of the nose, even if it prevents water reaching where it should, because these are things that usually happen a great deal, and if wudoo’ were not valid in the case of such matters, then the Prophet (blessings and peace of Allah be upon him) would have explained it, because it was not permissible for him to delay explaining at the time when such knowledge was needed.
To that – i.e., the issue of a small amount of dirt – Shaykh Taqiy ad-Deen ibn Taymiyah (may Allah have mercy on him) added every small thing that could prevent water reaching the skin, such as dried blood or dough on any part of the body. This was the view he favoured by analogy with dirt under the nails.
That also includes cracks that may appear on some parts of the body. End quote.
What is enjoined in the Book of Allah is to wash the face. Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! When you intend to offer As-Salat (the prayer), wash your faces…”
With regard to the area under the chin, it is not included in the definition of the face, so it does not have to be washed.
Al-Kaasaani said: Because the face is what is seen when looking at a person; whatever is not seen is not part of the face.
End quote from Badaa’i‘ as-Sanaa’i‘ (1/3).
See also the answer to question no. 129353.
With regard to the hadith mentioned in the question, there is a difference of opinion concerning it.
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: Regarding the hadith of Anas, it was narrated by Abu Dawood, but its isnad includes al-Waleed ibn Zarwaan, whose status is unknown….
It also has other isnaads from Anas, but they are da‘eef (weak).
End quote from at-Talkhees al-Habeer (1/86).
It was classed as saheeh by Ibn al-Qayyim in Tahdheeb as-Sunan, and by al-Albaani in Saheeh Abi Dawood.
Assuming that it is saheeh, the command mentioned in the hadith may be understood as meaning that it is mustahabb, so as to reconcile between this and other reports, because the majority of those who narrated reports about the Prophet’s wudoo’ did not mention running the fingers through the beard (to make the water reach the skin); if it were obligatory, he would not have omitted it from his wudoo’ (and would have done it every single time), and if he had done it in every wudoo’, then those who narrated descriptions of his wudoo’ would have mentioned it, and they would have made sure to describe how he did it in a clear way, or most of them would have done that.
For the ruling on running the fingers through the beard (to ensure that water reaches the skin), please see the answer to question no. 85031.
Determining what materials prevent water from reaching the skin is dependent on the nature of these materials. The scholars studied this matter, and reached the conclusion that materials are of two types:
The first type is material that has substance, i.e., it remains as a layer (on the skin), so when it is placed on the body it remains cohesive, separate from the skin, and can be removed by peeling or wiping it off.
The second type is material that has no substance, i.e., it does not form a layer, and as soon as it lands or is placed on the body, it loses its cohesiveness and it breaks down and is absorbed by the skin, and there is nothing left of it; rather only traces of it remain, such as colour and the like.
See the answer to question no. 240518.
Based on that, it may be said:
If the dirt has substance in the sense that it forms a layer on the skin to the extent that it prevents water from reaching the skin, then it must be removed when doing wudoo’.
If it is simply a matter of colour, or it had substance but it is very little and can be removed with water, then it does not make wudoo’ invalid.
And Allah knows best.