Both definitions are used and authors usually make it clear which one they mean. Wikipedia also points this out under the Alternative Defintion section.

The eigenvalues are just labels for the outcomes of the measurement. They don't necessarily have any physical meaning. See this answer for a more detailed understanding.

I'm not sure what $End(H)$ is but here is a proof that unitaries take density operators to density operators. Let $X$ be a positive semidefinite matrix with unit trace. We have to show that for a ...

Let us pick a basis for the Hilbert space $\mathcal{H}$ of dimension $d$ to be $\{\vert i\rangle\}$ and let $\mathcal{P}: \mathcal{H}\rightarrow\mathcal{H}$ be the dephasing map given by $$\mathcal{P}:...

A valid measurement consists of operators $\{M_k\}$ such that $\sum_k M^\dagger_k M_k = I$. You need appropriate normalization factors in front for both cases but since $P_i^\dagger P_i = P_i$ and $...