When is equilibrium analysis appropriate? What is a good rule of thumb to use to determine if equilibrium is the appropriate assumption?  I assume it is mainly to do with the time scales of the problem, meaning if the reaction time scales are very short, we could probably assume that we get to equilibrium very quickly.  Is there an accepted metric for determining if the equilibrium assumption is valid?
It depends on the context.

The accepted threshold is "when everything else is slow by comparison." For example, if you have a reaction that takes place in two parts, a Jones oxidation taking a secondary alcohol to a ketone and Diels-Alder (which is far slower), you can assume for modeling purposes that the oxidation will be in equilibrium whenever you're looking at how the Diels-Alder will interact with the oxidation.

By contrast, if you're reacting something like a $\ce{CN7}$ anion (which will react very very quickly, detonating with no provocation), the Jones oxidation can't be considered in equilibrium by comparison.