The sword icon indicates any city improvement allowing for vet ground units (Palace in this case), not only Barracks.
I'm against a bigger war in the next time. We need to get infra on our cities first. Shield costs of units go way up, no chance to keep up with the AI without production boosters. To be able to afford Forges etc everywhere, we need commerce buildings and so on.
And we won't have much luck with pointy stick research towards Literature, the AI usually avoids Drama, Poetry etc.
Believe me, we're way better off when we develop our core now, get up the needed small wonders (Heroic, FP, Training Camp), collect a nice stack of MWs, and strike later. RaR doesn't work with empty cities.
Btw, what are we building a Galley for? Can't enter Ocean anyway, so no suicide runs, and Trading Ships are better naval transports.
You're all aware that COnstruction allows for Mines? Obviously, this is another argument against a war now - the AI will have them all over the place already...