Can you choose not to have the electric vehicle charged by the battery, but only at times with cheap energy?

You can choose a strategy per asset (including charging stations). The strategies are described:

If, after the installation of the controller and battery, controllable equipment is added to the home by the residents, can the people themselves also add this equipment to the controller or does the installer have to do this?

As an installer, you are free to choose whether you share the local configuration page of the smart grid controller with the end customer. However, tinkering with the local configuration page without prior knowledge can lead to problems with the control.

Configuring the controller in terms of strategies is always done via the APP or web portal.

We now assume that the battery is controlled by the controller via a cable, could this also be possible via an API?

There are also systems that we control via API or webhooks. Depending on the manufacturer of the battery system, it is sometimes possible to do this remotely