The clock continues to work
Wake-up is also possible (but not with radio but only with alarm sound)
The display can be dimmed (either automatically depending on the ambient lighting) or manually or by time control.

The clock/alarm function also works without WLAN. The display can be either time-controlled or dimmed via a brightness sensor. Our clock is controlled via the brightness and works wonderfully.

As long as the clock was set before, this works without problems. I don't know what it looks like when the clock was turned off before, but I could test it this evening. The alarm function also works without Wi-Fi.

I don't think so, the time is obtained through the WLAN. The alarm clock is probably also triggered by it.