That's a great set of questions regarding safety features. To start out with, there overcurrent/voltage flags that can be configured to shut down the motor at arbitrary levels. So if, for example, the motor hits something and stalls (spike in voltage under "voltage mode"), it can detect it and shut off. This does need to be configured though and depends on the motor size.
Additionally, there is a switch input that can be configured as a homing or e-stop switch. It's attached to a pull-up, so a number of switches can be connected in parallel and if any of them are grounded, it will enter hard-stop (active bridge) mode. The switch logic is built into the motor controller and is independent of the wifi chip (esp8266), and as such there is no latency and the stop is immediate.
Finally, the wifi has a few safety features as well. While the web interface doesn't have auto-shutoff when disconnected (although maybe I *should* now that you mention it), the low level TCP and UDP connections that python uses do. There is a configurable heartbeat that will trigger a shutdown mode if there is a timeout. The idea is that, once you get the right configuration and motion commands figured out through the web interface, you'll most likely want to use python/etc. to integrate the motor commands into a larger application. That's where the "full power" of the security features (crypto chip) and safety features kick in.
Does that satisfy the question? I could go into more depth if you like.