My reading of it is that the app on the phone is used to configure the button, and the button from then on operates independently (which would be good in that it would allows others to use it when the original person isn't around). It's ambiguous, though, and could go either way.
Given that it would literally only have to connect to the wifi and do the API calls when the button is pressed, the power usage would be minuscule.
Given that it would literally only have to connect to the wifi and do the API calls when the button is pressed, the power usage would be minuscule.