Here are a few remarks to show that your Questions 1 and 2 are almost equivalent. (What the answers are is another matter, and I expect that to be somewhat difficult, as I'll explain below.) Of course, Question 1 is a special case of Question 2 in the OP's situation because, if a closed form is a scalar multiple of $\omega^2$ or $\omega^4$ on $M^{10}$, then that scalar multiple must be constant, and hence the form is parallel. What's interesting is that, in spite of appearances, having the Chern-Weil representatives of the Pontryagin forms be parallel when the holonomy is a subgroup of $\mathrm{SU}(5)$ is nearly the same as having these forms be scalar multiples of $\omega^{2k}$.

First, for Question 2, you can reduce to the case in which the holonomy acts irreducibly: If $(M,g)$ is a (simply-connected) Riemannian manifold whose holonomy preserves a parallel splitting $TM = V_1\oplus V_2$, then, locally, $g$ is a product metric and the formula for the total Pontryagin class $p(TM) = p(V_1)p(V_2)$ and that fact that the holonomy is the product of the holonomies acting on the two subbundles $V_1$ and $V_2$ shows that the Chern-Weil representatives of the classes in $p(TM)$ are parallel with respect to the holonomy of the product if and only if each of the two factor metrics has the property that its Chern-Weil representatives of the classes $p(V_i)$ are also parallel. By reduction, then, it suffices to classify the metrics $(M,g)$ for which the Pontryagin forms are parallel and the holonomy acts irreducibly.

Second, although you say you are mainly interested in the case of an $(M^{10},g)$ whose holonomy is (contained in) $\mathrm{SU}(5)$, you might as well generalize your question to considering the case of $(M^{2n},g)$ with holonomy (contained in) $\mathrm{SU}(n)$. Since, by the first remark, we can assume the holonomy acts irreducibly, that means, by the Berger classification, that the holonomy has to be either $\mathrm{SU}(n)$ or, if $n$ is even, $\mathrm{Sp}(n/2)$.

In the case the holonomy is $\mathrm{SU}(n)$, the only parallel forms of type $(2k,2k)$ are the (even) powers of the Kähler form $\omega$, so, in this case, having the Pontryagin forms be parallel *implies* having them be (constant) multiples of the powers of $\omega$. Meanwhile, for $k<n$ the only multiples of $\omega^k$ that are closed are the constant multiples. In particular, when $n=5$, if the Chern-Weil representative of $p_1(M)$ (respectively,$p_2(M)$) is a multiple of $\omega^2$ (respectively, $\omega^4$), then it is a constant multiple, and hence parallel.

In the case the holonomy is $\mathrm{Sp}(n/2)$, things are a little more complicated: In this case, the algebra of parallel forms is generated by $\omega$, $\phi$, and $\bar\phi$, where $\phi$ is a parallel holomorphic $(2,0)$-form. Thus, the parallel $(2k,2k)$-forms are constant linear combinations of the forms $\omega^{2(k-j)}\wedge\phi^j\wedge\bar\phi^j$ for $0\le j\le k\le n/2$. Since we only care about the cases $n\le 5$ for the OP's problem, we really only have to deal with $n=2$ and $n=4$ in this situation. When $n=2$, since $\mathrm{Sp}(1)=\mathrm{SU}(2)$, this case is already handled. When $n=4$, the only case to deal with is $k=1$, where there are two parallel $(2,2)$-forms, $\omega^2$ and $\phi\wedge\bar\phi$. However, in this case, because there is a $2$-sphere of parallel complex structures and because the Pontryagin form $p_1(M,g)$ will have to be of type $(2,2)$ with respect to *all* the complex structures of this family, one finds that, if the Chern-Weil Pontryagin form $p_1(M,g)$ is to be parallel, it must be a constant multiple of $\omega^2 + \phi\wedge\bar\phi$. (Here, I am assuming that $\phi$ and $\omega$ have been suitably normalized.) This multiple can't be zero because it is essentially the squared norm of the curvature; if it were zero, the metric would be flat, and the holonomy wouldn't be $\mathrm{Sp}(2)$.

Thus, the only case that you care about in which the two Questions essentially differ is the case of an $(M^{10},g)$ that is a product $(N^8,h)\times (\mathbb{C},g_0)$ where $g_0$ is the flat metric and $(N^8,h)$ has holonomy $\mathrm{Sp}(2)$ with $p_1(N,h)$ being a (constant) multiple of $\omega^2 + \phi\wedge\bar\phi$ and $p_2(N,h)$ being a constant multiple of the volume form.

Finally, there is the question of whether any of these irreducible cases can actually occur. Here, I don't have much to say. I do not know of a single example of even a local, non-flat Calabi-Yau metric for which the square norm of the curvature tensor is constant, even in complex dimension $2$, which is the very first nontrivial case. The problem gets more and more overdetermined as the dimension goes up, so it seems unlikely that such metrics exist, even locally, but I think that actually writing down a proof might not be easy.