Using thermodynamics to predict the acid-base character of fluoride ion in water
The textbook walks us through the interpretation of these enthalpies (they also caution us not to use enthalpies as sole criterion for spontaneity of reactions):



So they are asking whether water reacts with the acid HA, and whether water reacts with the conjugate base $\ce{A-}$. 

If we try that for the case at hand, we first consider whether HF reacts with water:

$\ce{H2O(l) + HF(aq)-&gt;[?] H3O+(aq) + F−(aq)}$

Here, water and fluoride are the bases "fighting for the proton" (reactants: fluoride has the proton; products: water has the proton). Fluoride wins (i.e. HF does not give away the proton) because it has the higher proton affinity, and the reaction does not proceed judged by enthalpy alone.

Second, we consider whether the fluoride ion reacts with water:

$\ce{H2O(l) + F-(aq) -&gt;[?] OH-(aq) + HF(aq)}$

Here, fluoride and hydroxide are the bases "fighting for the proton". Hydroxide wins because it has the higher proton affinity, i.e. this reaction also does not proceed judged by enthalpy alone.

So HF is not a strong acid, and fluoride is not a strong base. Another way of saying this is that hydroxide is a stronger base than fluoride, and hydronium is a stronger acid than HF.

In the textbook walk-through, they conclude that HA is a strong acid and - for a different reaction - $\ce{A-}$ is a strong base. In our case, HF is a weak acid  and fluoride is its conjugate weak base. If you try the same question with HCl, you will see it is a strong acid and chloride is not a strong base. (I don't understand the conclusion of the solution manual - they might have gotten confused because they know fluoride is a weak base, but their calculation together with the textbook argument merely shows that it is not a strong base.)

Now to your questions:


  
  Why does the text consider the second reaction with H2O? Doesn't F− act as a base in both reactions?
  


I think they ask "does HF react with water?", and "does fluoride react with water?". I don't know why the answer flips one of the reactions (maybe the solution manual wasn't written by the textbook author, and the original intent of the question got lost).


  
  If both reactions occur and F− acts as a base, then which one is dominant?
  


There are three equilibrium reactions, the two shown, and auto-dissociation of water. I don't know the kinetics of the three, but they are all fast and the three equilibrium constants are linked because if you subtract the two reactions you show, you get the auto-dissociation of water. For the equilibrium state, it does not matter which set of two you consider.


  
  If the first reaction is dominant, then how does it proceed given the low concentrations (1x 10−7 M) of H3O+ in neutral water?
  


Reaction rate constants vary by many orders of magnitude. The reaction of fluoride and hydronium is nice because the reactants are of opposite charge, so maybe it has a high rate constant. The reaction of fluoride and water is nice because water is the solvent, so it is present at high concentration and already at close proximity. I could not tell you which reaction is faster even if I know the concentrations of all the reactants.

Here is the table of proton affinities as a quick reference: