Is every deed in which the name of Allah is not mentioned rejected for its doer and without reward, as mentioned in the meaning of the hadith narrated from Aisha, may Allah be pleased with her: "Whatever in which the name of Allah is not mentioned is rejected"?

It appears you are referring to the hadith of Aisha from the Prophet, peace and blessings be upon him: "Whoever introduces into this affair of ours anything that does not belong to it, it will be rejected" (agreed upon). You seem to have confused it with the hadith: "Every matter of importance that does not begin with 'Bismillah Ar-Rahman Ar-Rahim' is cut short," which is weak. The meaning of the first hadith is that it is not permissible to worship through an act unless there is evidence to support it; otherwise, the act of worship will be rejected and its doer will be sinful.