Why doesn't nitrogen react with oxygen at room temperature? I already know that nitrogen's triple bond requires a lot of energy to be broken, so the activation energy is very high, but the book also lists the endothermic energy change as a reason. 


Does the positive enthalpy change have something to do with the reaction not being spontaneous i.e. due to positive free energy? 
Is there a link between $E_a$ and free energy when deciding whether a reaction will occur?
Yes, you are right, the reason primarily looking at it from a theoretical point of view is the high bond dissociation energy of $\ce{N#N}$ 

Let's look at this aspect from every point of view


Theoretical- High bond dissociation energy of $\ce{N#N}$
Kinetics- We have the Arrhenius equation given below, which states the relationship between the rate constant and activation energy of the reaction. 


$$k = A \mathrm e^{-E_\mathrm a/(R T)}$$ 
So essentially, this relation shows that if activation energy is higher, then the rate is slower. However, if we increase the temperature sufficiently, then the reaction does start to happen significantly. (Hence, oxygen and nitrogen do react when lightning strikes)


Thermodynamics- We have the very famous relation of thermodynamics 


$$\Delta G= \Delta H-T \Delta S$$ So now lets again look at each term here


$H$- Enthalpy of the reaction, it is very high, as we discussed before, the bond dissociation energy of $\ce{N#N}$ is very high. 
$S$- Entropy change won't be any significant as the reactants and products both are gases, so this is not an entropy-driven reaction.


Thus
, in every way possible we can conclude that nitrogen and oxygen do not normally react at room temperature.