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Last time I checked, the entire function $f \colon \mathbf C \to \mathbf C \colon z \mapsto a\exp(iz)$ was transcendental for every non-zero $a \in \mathbf C$. So, taking $\alpha = -a = 2\pi$ yields a negative answer to your first question; taking $\alpha = a^{-1} = 2\pi$ yields a negative answer to your second question; and taking $\alpha = a = 2\pi$ yields a negative answer to your third question.