Well, the Hermitian adjoint is defined, at least in finite-dimensional Hilbert spaces, as the complex conjugate transpose. Scalars you can think of as 1-dimensional vectors. Transposing a scalar doesn't change it, but the complex conjugate part does (if it's complex, of course).

*Hence, the Hermitian adjoint of a scalar is just the complex conjugate.* The fact that I'm right-multiplying by the scalar is unimportant, as scalars can pass through vectors no problem (that is, scalar multiplication of a vector is commutative).

I think you're confusing two different proofs.

1. If you're trying to show that the eigenvalues of an Hermitian operator are real, then you play around with

not

The reason is that you have to know that

which is true because eigenvectors, by definition, are nonzero.

2. If you're trying to prove that the eigenvectors of differing eigenvalues are orthogonal, then you play around with

where the eigenvalues of the two eigenvectors there are different.

I would definitely prove that the eigenvalues are real before proving that the eigenvectors of differing eigenvalues are orthogonal.