Shouldn't it be "~te MO ii desu ka"?
I believe you can emit the MO to make it more casual... could be wrong though
You're correct, however, this app teaches you how to speak politely so it should teach you to say もいい
Wouldn't it be a better translation 'is it ok if I turn on the light'?
What is the いい all about?
It means good. The speaker is asking if it's (literally 'good') on to turn on the light.
Why is the 「です」 there?
...Why wouldn't it be?
Maybe because the folk wants to speak in a casual manner?