When does a male cousin's daughter inherit? And what is her share if a man dies, leaving a wife and one daughter?

The paternal cousin (female) does not inherit from her paternal cousin (male) because she is neither among the ashab al-furud (sharers) nor among the 'asabah (agnates). Therefore, for someone who passes away leaving a wife, a daughter, and a paternal female cousin, the wife receives one-eighth due to the presence of a lineal descendant (the daughter), and the daughter receives one-half. The remaining portion is taken by the daughter by way of radd (return), and the paternal female cousin receives nothing. The matter of inheritances is extremely serious and complex. It is safer not to rely solely on this answer, but to refer the matter to the Sharia courts to verify the heirs, as there might be an heir discovered only after investigation, or there might be bequests, debts, or other rights.

And Allah knows best.