Can creatures create things from nothing with the permission of Allah Almighty, like Jesus (peace be upon him) when he breathed into the bird and it became a bird with the permission of Allah? And does one who claims this commit an act of disbelief?

Creation ex nihilo is exclusive to Allah alone; no one else is capable of it. The Quran has shown the falsehood of the divinity of any other than Allah, for Allah is the Creator of all things, the One, the Subduer. Jesus, peace be upon him, did not create a bird ex nihilo; rather, he transformed clay into the form of a bird by Allah's permission. The term "creation" can refer to bringing something into existence from nothing (ex nihilo) or to shaping and forming (estimation/predetermination). In the case of Jesus, what is meant is shaping and forming, not creation ex nihilo. The attribute of "the Creator" is exclusive to Allah and is not permissible for anyone else. Whoever claims to bring something into existence ex nihilo is a disbeliever.