You can manually switch that if you prefer though. Any unit that's placed in the "general field" on the pre-battle screen will be the commander. Rarely worth it though, but that's another matter.
About the kings and princes, the main problem with using the king in battle is that the "distance to king" loyalty factor will always be maxed out (aka your king is trapped on Ireland with no porth) when the king attacks. So it's devastating to do it when your kingdom is large, because it will cause massive rebellions.
The princes don't have this disadvantage though, so use them freely. Placing them all in the same army unless you need to is quite disadvantegeous though. The non-commanding hiers don't aquire traits as easily inside a stack and on the battlefield. They can also easily get killed by accident (they have more hit points than the lieutenant, but less than the commander) as the game doesn't inform you of thier death. And losing your entire royal family in one loss is devastating to say the least.
Finally I can say that you don't need to command the royal family into battle at any point to breed them to have 7-9 in all stats, but it's easier (and "accidental" deaths of lousy elder sons helps as well).
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