is my 13.9-year estimate for how long it takes for a population-reducing event like the one I described realistic? Why or why not?
First, it all depends on this statement of yours:
1% of Earth is rendered uninhabitable by the war.. 71% of earth is water, and 29% is surface. Roughly 57% of surface area is uninhabitable, which means you are left with 29%*43% = 12.47% of area as habitable. Even within this area, you will find clusters of populations centered around major cities and major cities around major rivers.
Assuming you meant 1% of earth's area, sure, there is already a lot of area that is not habitable, so your rate is achievable.
But the moment you meant to subtract 1% out of the 12.47% above, you will find effective habitable area available has decreased by ~8%. If this decrease happened in areas where major cities are located, or major centers of forests are located, you have seriously hurt the economy and the environment respectively, which will make a recovery seem more distant.
Second, you are assuming people still want to breed like rabbits after that brutal war. But why should things normalise so soon? Statistically, roughly 1 in 7 persons died. Assuming an average person knows 150 people, an average person will know 21 people who died during the war. You haven't noted that only males die off in majority so the average sex ratio has been impacted, or any other such hypothesis (which happened during the world wars). So, the psychological effects of such a drastic war will be manifold, and a chunk of people may actually become disinterested from starting families, reinforcing a trend that is already seen in current era.