Definitely the parent's responsibility to fit a stairgate if there is any risk of their child wandering around at night. Even with a working light, children can still stumble and fall when sleepy.
OP, without seeing the house it is hard for us to judge how serious your concerns are but to be honest, you seem to be expecting too much from your landlord.
And I'm still intrigued by the nature of the 'flammable substances' on the bathroom ceiling...
'Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you, but for the formation fo the first link on one memorable day'. Charles Dickens, Great Expectations