Number and function of voltage-independent ion channels
Those would be referred to simply as 'ion channels'. Voltage-dependent channels should get special mention.

This link from the Blue Brain Project should prove useful: 


  http://channelpedia.epfl.ch/ionchannels/



A query for 'voltage-dependent' gives me results from 58
channels. According to the original
reference
there should be at least 187 channels in this database (out of the
estimated ~500 ion channels expressed in the brain). That should
give you at least the order of magnitude.

An additional resource can be: https://icg.neurotheory.ox.ac.uk/.
'Non-voltage gated' channels are used to depolarize and
hyperpolarize the membrane. That is as much as specific the answer can be when talking about all of the ion channels. Their specific compartment localization makes a big difference, though. Those that are located in synapses have different functions compared to those that are non-synaptic, for example.

No, voltage-dependent channels are not
solely found in the axon. That is what makes dendrites active
compartments as well. See this very good
review
and my
answer
to
How does a neuron change as you learn?.
In brief, voltage-gated channels facilitate the integration of
inputs on the dendrites independently from the integration at the
level of the soma. This is thought to extent the information
processing capacity of the
neuron.


PS: A lot of your questions suggest that you would be interested in the NEURON simulation environment, as previously suggested to you by @bryan-krause. Using this tool could save you some trouble of trying to re-invent the wheel, as you can start off by using a pre-existing model from ModelDB.