I would like to see the compilation results after hitting C-c, however, AUCTeX always asks me to additionally hit C-l to swich to the log window and automatically scroll down. I would prefer C-c to include this step.
Since AUCTeX 11.88 there a new errors and warnings overview (see also https://tex.stackexchange.com/a/185823/31416). To automatically open it after running a TeX engine set the variable
TeX-error-overview-open-after-TeX-run to t. To do that, add the following code to your init file:
(setq TeX-error-overview-open-after-TeX-run t)
Here is how the overview looks like:
You can open the error overview whenever you want with M-x