I specified it in my post above (see "EDIT"). In short my question was, why $g_p(t)(X,Y) := \tilde{g}_{f^{-1}(p)}(t)(df^{-1}X,df^{-1}Y)$ is well defined for $t>0$. But this was answered by Robert, so I updated my post.

@Otis Thank you for the patience to summarize these results. So given a manifold of bounded curvature I could use any Ricci flow solution $(\tilde{M},\tilde{g}(t))$ with $\tilde{g}(0)=\tilde{g}$ on the universal cover $(\tilde{M},\tilde{g})$ to obtain a solution on my original manifold $(M,g)$. (The question, how this is done, still remains.)

Thank you, somehow I overlooked that fact. So there is no need for the last part about Kotschwar's result and I will delete it.

This variant of Shi's estimates is indeed very interesting - thank you for the hint. But if I had read this result first, my question would have been the same: What do I need for bounded derivatives of the initial curvature? According to the answers above I now think a closed manifold with smooth metric will do.

First of all thank you for the answers. I was aware of the equivalent formulation to my second question and just wanted to give my motivation for it. But now I seem to misunderstand some (presumably trivial) point here. Aren't these functions $g_{ij}$ always smooth? I thought, this was implied by the definition of a riemannian metric.