quantum mechanics - Sine function as a momentum eigenfunction

I'm confused as to whether the sine function can be technically considered an eigenfunction for momentum operator.

Once the sine function is decomposed, it can be decomposed as a linear sum of two eigenfuntions for the momentum operator since $$\sin(kx)=\frac{1}{2i}\left[e^{ikx}+e^{-ikx}\right] \, .$$ Applying the momentum operator on each of these functions gives $hi$ and $-hi$ for the momentum values. However, applying the momentum operator on $\sin(kx)$ itself clearly shows that it is not an eigenfunction one cannot recover the original sine function after applying the operator. Why is sine not an eigenfunction despite the fact that it is a linear sum of two eigenfunctions?

Also, for a particle in a box, one can use its wavefunction (a sine function $\sqrt{2/L} \sin(n\pi x/L)$) to calculate the average momentum as 0. If sine is not an eigenfunction for momentum, how can we calculate the average for the momentum observable?