$pi_2(T vee mathbb{C}P^2)$ and action of $pi_1$ on $pi_2$

You're right about the universal cover, so you should try to describe it explicitly to understand its $H_2$.

Then for the action of $pi_1(X)$, note the following result:

Let $X$ be a nice connected space with universal cover $tilde X$. Then there is a nice action of $pi_1(X)$ on $tilde X$. This induces an action on $pi_2(tilde X)$ : with the isomorphism $pi_2(tilde X)cong pi_2(X)$, this is exactly the action of $pi_1(X)$ on $pi_2(X)$

Note that this is not well stated : indeed homotopy groups require basepoints, and the action of $pi_1(X)$ on $tilde X$ does not preserve them, so as such, the action on $pi_2(tilde X)$ not well-defined.

But $tilde X$ is simply-connected, so $pi_2(tilde X, x)$ and $pi_2(tilde X, y)$ are canonically isomorphic, via any path $xto y$ in $tilde X$.

Therefore if you're precise you can state this result correctly, and with the correct statement it becomes almost obvious.

But then you can compute this action on $pi_2(tilde X)$ via the same action on $H_2(tilde X)$ by Hurewicz, and with a geometric understanding of $tilde X$ (and of why its homology is what it is), you'll get a geometric understanding of the action on $H_2(tilde X)cong pi_2(tilde X)cong pi_2(X)$