If one is flexing a muscle, for any given motoneuron involved, how many action potentials occur per second to keep the muscle flexed? Also, will the axon terminals in a given motoneuron ever run out of neurotransmitters to release if they are constantly undergoing action potentials (say in the situation outlined above where a muscle is being kept flexed)? Or is there always enough reuptake to ensure that this could never happen?
Second question: won't happen in the healthy individual/inapplicable.
Anatomically speaking, muscles don't flex, they contract. Joints flex. And muscles never fully relax to the point of zero contraction/zero firing. (The flexion of a joint can be achieved passively, even when the animal is dead.)
Even at rest, muscle is kept in a constant state of at least slight tension, the basic tonus. Skeletal muscles are usually set up in antagonistic pairs, with one flexing and one extending a joint. A specific balance of agonist and antagonist tonus implements a stable position of the joint, and stable positions can be achieved near-permanently.

First question: depends on effort.
Muscle firing during increased, active contraction is a function of effort, as muscle contraction is increased via rate coding. The stronger a muscle is supposed to contract, the higher the motor unit's firing rate has to become. (I should probably mention the size principle here...)
Motor neurons are reported to reach firing rates of a few tens (up to 50) of hz during high effort.

A question you're possibly interested in: for how long can extremely high firing rates be sustained? Or: is muscle failure under sustained extreme effort, corresponding to high firing rates, sometimes a result of motor neuron drop-out rather rather than e.g. ATP depletion at the muscle itself?
Note first that muscle force is greater for eccentric and isometric activity (ie., release under control or static tension); voluntary contraction is a bit weaker. From another perspective, if a muscle is electrically stimulated externally, it is still able to exert force far beyond the point where the subject cannot voluntarily uphold the tension. This means short-term fatigue following high tension is primarily neural.
I do not know what precisely causes this fatigue (nor do I think the experts have uncovered it yet), but the limiting aspect does not seem to be neurotransmitter re-uptake, or even metabolic processes in general. As Gandevia writes:


  intrinsic moto neuronal properties, reflex inhibition and disfacilitation, Renshaw cell inhibition, and insufficient drive from supraspinal sites may all contribute to the decline in moto unit firing rate in fatigue


Also see this.