How old is Islam? Non-Muslim textbooks will trace Islam’s origins to the Arabian desert around 600 AD (or CE if you prefer) with the life of the Prophet Muhammad (pbuh) and the establishment of the Islamic Ummah.
Muslims themselves are far more ambitious, tracing their religion to Creation itself, for in the beginning, God created Islam. Since then He has consistently revealed His true religion to the prophets, the last of whom was Muhammad. Thus, for Muslims, Muhammad was not the first Muslim. This honour belongs to Adam. Abraham was a Muslim, Moses was a Muslim, David was a Muslim, Jesus was a Muslim.
The followers of these earlier prophets (Jews, Christians) eventually apostatized from the true message. It was therefore restored in its fullness and finality to Muhammad.
Regarding David the Muslim, Islam cleans up his story considerably. David was a prophet and prophets are righteous therefore David was neither an adulterer nor a murderer. The Bible, it seems, is only true insofar as it is translated correctly.