Does a neuron ever generate an action potential without stimuli?
Neurones fire when they're sufficiently depolarised past their threshold. In the sensory system this is in response to an external stimuli which sets of a chain reaction. Then motor neurones may relay signals back and in this case the signal or initial depolarisation is occurring in the brain. However this is still caused by a stimuli even if indirectly. If we take an action it is generally because we thought which action we should take because of some sort of stimuli e.g. we are cold so we put a coat on. 

But anything which depolarises neurones can set of an action potential. However there's always something which causes it to depolarise, perhaps a drug or a stimulus or another neurone. If any of these don't count as a stimuli depending on your definition, there's your answer. 

Or perhaps if you consider cardiac muscle which can intrinsically depolarise. Not a neurone but there's an action potential. They just naturally depolarise over time and action potentials are set of regularly with no stimulus required. 

As for learning, where the structure of neurones can change so they connect to new neurones or not to old neurones or vary in the strength of their connection.