Let $A$ be a $C^*$-algebra and $B=pAp$ for some projection $p\in A$. Let $N=A^n q$ be a finitely generated projective (left) $A$-module, where $q$ is a projection in $M_n(A)$. Then there is a $B$-action on the tensor product $$T=(pA)\otimes_A N.$$ from the left.
Question 1: Is $T$ a finitely generated projective $B$-module? If so, what would be a projection $r\in M_k(B)$ such that $T=rB^k$?
Now let $i\colon M_\infty(B)\hookrightarrow M_\infty(A)$ be the inclusion.
Question 2: Are the projections $i(r)$ and $q$ similar inside a large enough matrix algebra over $A$?
Some context: This question is motivated by this question and the comment on it by Alain Valette. In particular, $pA$ is supposed to be an imprimitivity bimodule between $B$ and $A$, and Q2 is basically asking how to show that the assignment $N\mapsto T$ gives the inverse to the map $K_0(B)\to K_0(A)$ induced by $i$.