Regarding a consumption aggregator: How do I differentiate under the integral sign?
Using your formalism above, you can think (heuristically) of the integral as $$ \sum_{i=1}^nC_t(i)^{(\epsilon-1)/\epsilon} $$

If we differentiate this with respect to $C_t(j)$, we get

$$ \frac{\epsilon - 1}{\epsilon} C_t(j)^{-1/\epsilon} $$

Which is exactly what we needed. To do this rigorously, you need a notion of taking derivatives on function spaces. Look up the Gâteaux and Fréchet derivatives.