1) The Western Part of Scotland belongs the jure to my kingdom but is governed by a duke who is independent. Is this not a valid casus belli towards him?
No, but you can ask him to become your vassal. In order to do this you need to prove your power to him and make him happy. Proving power could be done by attacking England, which is currently under attack by Norway and some other people or you decide to press your claim on the northern part of Scotland, which immediately occurs when starting to play as the King of Scottish Skirts. Winning one of these battles will demonstrate your power and he may like to join you as a vassal.
2) One of my vassal was a duke and had 4 counties. As he was too much powerful i decided to revoke his title of duke. (i had the necessary crown authority to do it). After some days of thinking, he refused to do so and so a war began. I won, i imprisoned him. So i could revoke title, as i did it, my second son became duke. Is there a way to decide who become duke? Would have been different if i decided to revoke one of his counties?
Erm, if you revoke the title, the title belongs to you, so you have to decide, who you want to have it. I do not understand, how your second son could get this title. Also it is a very dangerous thing to give your temporary heirs titles because they will fight each other for your king title as soon as you die. Always give your first heir land, so that he can gain prestige but not the other ones.
3) where can i see anybody's claim?
- If you look at the character screen of one person, you can see this persons titles and claims.
- If you want to know, if you can declare war one someone because of a claim, try to declare war on this person or his ruler and look at the list of possible claims.
- If you want to know, if there are any people outside of your dynasty claiming something, click on the flag symbol and then on the button that should say something like "claimants". You can invite those people to your house, if they display a green thumb up.
4) about change of laws: what does depend the speed of votes becoming "yes"? From the relationship with the vassal? How does the vote system work?
Yes, if the relationship of the voters is 50+ towards you, they will agree.