I cannot answer question a) as I don't know why they are fighting each other in the first place...
b) Daimyo (Annuity): It says that it may switch places with the Shogun even if the Shogun is "on the prize stack of any
player". How can this be? Will I became the Shogun and give my Daimyo card as a prize in return?
If you choose to swap cards then you will take the Shogun and give Daimyo in return. Both cards will end up in the same place where the other was.
The Shogun may be defeated in battle and thus end up a prize stack. See the example under "1. Attacking another Lord".
c) Can the ninja kill (assasin) the shogun? This means that if you are the ninja your aim is to lay low until every other player has played his disk and then atack the hiest in rank to get the prize (his lord in points), right?
Yes, the Ninja can kill anyone except another face-up Ninja.