Does the C-7 Hub generate any kind of alert if it thinks its internal temperature is too high?
I do not know of one but it is easy to construct one using the Hub Info driver and a simple rule or notification.
Yes, it does send out an alert if you have Pushover. This I have recently experienced. Other than that, the temperature is shown in the logs.
EDIT: Turns out that the high temp alert is not from HE itself, but rather from an app I installed, so you can try that: [BETA -Release Candidate] Hub Information Driver Aggregation App