How do firing patterns arise from the activity of many ion channels?
Typically the activities of the ion channels are not called firing patterns as in neuroscience we refer to "firing" when we mean the elicitation of action potentials (spikes) but yes: Whenever an AP was fired a sufficient amount of sodium channels had to be open and therefore I thing your reasoning is correct. In other terms what you are saying is that the effective channel conductances change during an action potential.

On your conclusion: In the regime of natural parameters the time scale of firing pattern 1 mostly depends on the time constants of the voltage gated ion channels and not so much on the absolute number of channels (especially if all the conductances would scale equally). 
In the world of Hodgin-Huxley like coupled- and nonlinear-dynamical systems the voltage does not scale strictly linear with increasing the participating max. conductances. The interplay of max. conductance and temporal gating dynamics resulting in the effective conductance itself depends on the voltage and there are nonlinearities. So the correlation between the number of channels and the AP rise time is not expected to be strictly linear.

See the figure for a little example: I tested my reasoning by increasing the max. conductance for the sodium channel of a simple HH model and checked the max. steepness of the voltage (normalized by peak voltage).

Hope that answers your question(s).