Direct Sums in Learning Theory

For a class $C\subseteq Y^X$, let $C^r$ be its $r$-fold product, acting coordinatewise on $X^r$. For classes $C_1\subseteq Y_1^{X_1}$ and $C_2\subseteq Y_2^{X_2}$, write $C_1\otimes C_2$ for their product. Determine the following quantities tightly.

1. For a marginal distribution $D$ on $X$, can the fixed-marginal learning-curve bound $\varepsilon(n\mid D^r,C^r)\leq r\varepsilon(n\mid D,C)$ be asymptotically improved for some $C,D$?

2. How does the uniform-convergence curve $\varepsilon_{\rm UC}(n\mid C^r)$ scale with $\varepsilon_{\rm UC}(n\mid C)$ and $r$?

3. How does the agnostic PAC learning curve $\varepsilon_{\rm agn}(n\mid C^r)$ scale with $\varepsilon_{\rm agn}(n\mid C)$ and $r$?

4. If $k_i$ is the minimum list size for which $C_i$ is list PAC learnable, what is the minimum list size for $C_1\otimes C_2$ as a function of $k_1,k_2$?

5. Ask the parallel question when $k_i$ is the minimum list-compression size.

6. For $\dim$ equal to Graph, Natarajan, Littlestone, or Daniely--Shalev-Shwartz dimension, how does $\dim(F\otimes G)$ scale with $\dim(F)$ and $\dim(G)$?