A Father to His Men - Most of the kings do this to some extent (with the Green King being a mother to her men, obviously) but none on quite the same level as the Black Prince Seraiah, who actually refers to his soldiers as "my child."
Armchair Military - Averted in the kings, but many commanders within the various armies are this.
Attack Animal - The Blue Army's clockwork, the Black Army's specially bred soldiers, and (though treated more as allies then weapons) the Red Army's wyverns. The Blue Queen can qualify as well with her numerous cybernetics.
Forever War - Part of the founding principle behind the Red Army was that their king found war to be really, really fun and wanted to fight forever.
General Ripper - Many of the Kings, especially Red and Black. The Blue King comes close simply because as a machine he has no sense of morality; of course he also has no emotion meaning he can't enjoy killing either.
Green Rocks - The Blue Kingdom's magic crystals, even if they're not green. (They're light blue.)
Medieval European Fantasy - Played straight and averted. While Chess World contains many aspects of Medieval Europe most of the kingdoms are based on other civilizations, none of which are European.
The Musketeer - Blue soldiers' spears can be used to fire energy bolts, making them effective short and long range weapons.
My Country, Right or Wrong - A lot of citizens of the Blue, White, and Black Kingdoms. Isaac in particular despises his kingdom, but will still fight to protect it because he believe the White Kingdom is worse.
No One Gets Left Behind - The Black Army, especially Seraiah's unit. The Blue Army is a lesser example as, while Deep Blue considers most of his troops expendable, he will never allow the kingdom's technology to fall into enemy hands and will make them self-destruct first.
Normally, I Would Be Dead Now - The queen unit in Seraiah's faction. Subverted in that he is dead by the end of the fight, but keeps going anyway.
Power Equals Rarity - Averted and played straight. Most powerful weapons are mass produced; every White paladin carries a blessed sword and energy blasters are so common in the Blue Army even the pawns have them. On the other hand, in most kingdoms the really powerful weapons are generally reserved for the kings and queens, to ensure they'll always be a step above their followers.
Red Right Hand - Vampires look human except for their fangs and gold eyes with red sclera. Seraiah's eyes are black and red. Ironically the true villains of the series, the Anti-Paladins don't use this trope.
Schizo Tech - Most of the world is high fantasy/medieval, but then there's the Blue Kingdom's clockwork, and the Black Kingdom's genetic experiments.
Self-Destruct Mechanism - The Blue Kingdom's clockwork do not actually have Self Destruct Mechanisms. Rather, the Blue King has direct control over every piece of magic crystal on the planet and can cause them to detonate with a single thought.
Averted in the Red Kingdom's history: the Red King, betrayed and reanimated as a vampire, has his former king utterly at his mercy and every reason to spare him, but instead murders him and everyone else in the capitol city.
Subverted in the Black Kingdom's one and only conflict with the Blue Kingdom. After stealing some of the Blue Kingdom's magic crystals and trying to reverse engineer them they draw the wrath of the Blue Army. They prepare to fight back against them only to have the Blue King himself appear...and detonate all of the stolen crystals, turning the facility into a smoldering crater.