What is the role of non-synaptic plasticity in learning and memory? Is non-synaptic plasticity (e.g. changes in the spike threshold) believed to play a large role in learning and memory? If so, which roles are these effects believed to play?
The answer is yes. Non-synaptic plasticity exists as you can read in "Non-synaptic neuronal mechanisms of learning and memory in gastropod molluscs" by Benjamin et al. and "Compartmentalization of Non-Synaptic Plasticity 
in Neurons at the Subcellular Level" by Nikitin and Balaban.

But the effect of hormones such as cortisol may also have an effect. 

Non-synaptic mechanisms of regulation and plasticity play a very important role in the brain and its ability to learn in many aspects and at different scales. 

A single cell can suffer internal modification that will modify eletrophysiological properties. 

Variations of concentration of elements in the CSF also have an effect. 

Glia also plays an important role. 

So, the brain is a very complex system with a lot of interdependent mechanisms of adaptation, regulation and plasticity. Synapse is just one aspect of it.