Questions about Sonnenkönig Eco 700

That is exactly how it is.

If the "window open" symbol is visible, press and hold the "-" button until you hear the sound, then it will heat up again. Mine has been working perfectly for years.

No remote control came with mine, I have two of these panels as well as two of the more expensive ones in the range (Elegance 500/700). There is a small box on each which you use to control it, or you can use the app. I have found the app pretty good, you can set timers, automations, scenes, etc. I think it will be sufficient for your purposes. 4 heaters seems a lot for 40m2, I guess it depends how warm you want it and how quickly.

It is certainly the dimensions: Eco 115 x 60 x 5.6 cm, Elegance 120 x 60 x 5.6 cm. At the same time, the Eco has a white aluminium surface and the Elegance a white glass surface.

I don't think there's enough room for a bracket on a wall: https://static.digitecgalaxus.ch/Files/3/8/5/9/6/6/1/2/21221102_ECO20700_Produktblatt.pdf

The element already comes with spacer bars that are permanently mounted on the panel. The distance to the wall is approx. 2 cm. A towel can be hung over it, but I think the panel will be too hot for the towel.

Yes according to the website for wall and ceiling mounting.

Yes, it is quite basic to install it on the wall. I installed it on the ceiling, so two options are available. Wi-Fi controlling and setting schedules is OK.