When does your commander also participate in the battle?
For scenario 1, the answer is "Maybe".
Is your knight-commander also part of the army fighting? Then, yes. Otherwise, no.
It is entirely possible to have a knight who is a commander but is not participating in the battle as a knight. Probably most commonly occurs if you lose a knight early in a war and don't re-raise your troops from a rally point -- you can appoint any courtier as a commander, and if a courtier is an unraised knight, they will not participate in battle.
It is also possible for a knight to lead an army while being physically present in another army. How? No clue. Maybe really good messenger hawks.
For scenario 2, the answer is "No".
Only knights fight in battle, and since the player character is never a knight, they will not fight. However, Commanders can be wounded, slain, or captured in battle even though they don't fight -- being present is enough.
This is true even in scenarios without knights at all - in which case the message given is "Commander so-and-so was wounded by an enemy soldier" rather than the named knight.
The chance of this is partially based on Prowess, so it's still better to have high prowess if you're leading troops personally.
For scenario 3, the answer is "No".
As mentioned in scenario 2, only knights fight in battle, and since the player character is never a knight, they will not fight. (They're still in danger, however, and having a high prowess helps mitigate that)
- Yes it matters and he does, at the end of the battle you can see how many casualties each knight caused in the knights tab of the battle overview.
- Yes, and he can also be injured or killed in the battle
- Prowess indicates how good your character is at fighting, e.g. how his chances are winning an individual encounter. However the martial skill determines how good they are at leading a battle. It could be that your PC character is not a knight in your army because of low martial skill.