Is it permissible to be named Noah? And is the story that his name was "Tashkur" and then he was named "Noah" due to his excessive wailing sound or unfounded?

There is no authentic text that specifies the reason for Noah (peace be upon him) being named as such. The most that has been reported is the saying of Yazid al-Ruqashi: "He was only named Noah (Nuh) because of how long he wailed (naha) over himself." The name Noah is the name of a prophet who is among the 'Ulul Azm (Prophets of Strong Will) among the Messengers. It is commendable to name children after the prophets, as the Prophet (peace be upon him) named his son Ibrahim, and the Companions also named their sons after the prophets.