If the Jews believe that Moses was the greatest prophet and that his prophecy is eternal, do they have Aḥādîth (records about his day-to-day doings) and do they follow his Sunna (daily customs)? 

This is a very important question because it touches on an essential difference between the way in which Jews and Muslims “believe” in prophets.  Muslims, Christians and Jews all believe that a prophet is someone who receives messages from God Almighty. It is a religious obligation, once someone is established as a true prophet, that we study his message and follow his prophetic instructions. 

However, other than Muslims, most Jews do not believe that prophets are “Ma`ṣūm”; immune for transgressions or sins. Like Jews, Muslims also believe that prophets are fully human and not in any way divine (God forbid), but they still believe that a messenger of God is an “insān kāmil”; a perfect person, ranked higher than other mortals. Different from Jews also, Muslims often have the name of their prophet written in beautiful calligraphy, and attached to the walls of their homes, next to the name of God. To talk bad about a prophet is seen by Islam as a form of blasphemy and a capital offence. 
 
All this is different from Judaism. While belief in Muhammad is part of the Islamic creed: “Allah is the only God and Muhammad is His messenger”, the Jewish equivalent only mentions God: “Hear o Israel, the Lord is our God, the Lord is One.”
 
In addition, while Jews value and revere the message of the prophets, they are not seen as infallible. The Torah and the other prophetical Scriptures describe not only the virtues of the prophets, but also their shortcomings and even their sins. Even Moses, our greatest prophet and teacher, once did not closely follow God’s clear instructions, and as a punishment he was not allowed to enter the Promised Land. Similarly, the Holy Books of the prophets tell us about transgressions committed by David, Solomon, Jonah, and others.

Jews believe that the stories of the lives of the prophets are written in our Holy Scriptures so that we can be inspired by what they did right, and we can also learn from their mistakes. Furthermore, and very importantly so, we look for guidance not in the conduct of our prophets, but from the instructions of the Lord our God. Prophets were chosen for a task, namely to bring the words of God to their people. But they are still vulnerable and fallible, just like we are.

Perhaps we can explain this with a parallel example. It is known that the Israelites are called “the chosen people”. That does absolutely not mean that Jews are by nature better people than others, or that they are God’s favorites. Jews believe that being chosen means that God chose them for a task, namely to receive and practice the Torah, to bring monotheism where there is idol worship, to bring light where there is darkness, knowledge where there is ignorance, and righteousness where there is suppression. But this doesn’t mean that Jews always succeed in that and that they are better people than others.  Sometimes it seems that God chose the least suitable people for this task, in order to prove His power, as it says in the Torah: “The Lord did not desire you and choose you because you are the most numerous of all people, for you are the least of all peoples” (Deuteronomy 7:7). Like anyone else, Jews make mistakes, they need to ask forgiveness, and learn to improve themselves. In the same manner, when God chooses someone to be a prophet, it does not mean that this person is without mistakes.

The way that Moses conducted his daily business, was not necessarily better than the habits of other people. For that reason, there are no aḥādîth about how Moses dressed himself, what types of food he preferred, if and how he trimmed his beard, or other mundane things[1]. Even the details of purely religious commandments such as prayer and Shabbat observance were largely interpreted by the Sanhedrin, a council of 70 wise men that Moses was commanded to appoint[2] and which continued to exist for many centuries. It is on the decisions, interpretations and legislation of this council, and not on the examples of Moses and the other prophets, that the Talmud and Jewish practice is based.

[1] Although, starting in the Talmudic period, there are such records of later religious sages.
[2] Exodus 18:17-26 and 19:7; Numbers 11:16-17 and 11:24-25; Deuteronomy 1:13-18.