Let A be a finite dimensional semisimple algebra over $\mathbb C$, e a primitive idempotent of A, and P any idempotent of A.
Consider the semisimple algebra PAP. I would like to prove (or disprove) that (i) eP is 0, or (ii) for some $N \in \mathbb C$, N eP is a primitive idempotent in PAP. Any idea?
This is false. For instance, if $A=M_2(\mathbb{C})$, then you could have $e=\begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}$ and $P=\begin{pmatrix} 0&1 \\ 0&1\end{pmatrix}$, and then $eP=\begin{pmatrix} 0&1\\ 0&0\end{pmatrix}$ is not a scalar multiple of an idempotent.
What you can say is that $A$ is a product of matrix algebras, and $e$ is a rank $1$ projection on one coordinate and $0$ on all the other coordinates. This means that $eP$ will be $0$ on all but one coordinate and have rank at most $1$ on the last coordinate. A matrix of rank $1$ is either a scalar multiple of a rank $1$ projection or is a nonzero nilpotent matrix, but either case is possible for $eP$.