Your State Diplomacy (your leader's diplomacy, plus your chancellor's diplomacy, and some contribution from your spouse) determines a bonus to relations with all other characters in the game. It's very helpful. If you raise your children yourself (and you should in most circumstances), always choose event options that give them diplomacy traits. Children now get the same education type as their guardian however, so if your leader isn't a Grey Eminence (just about the best trait), Charismatic Negotiator, etc then consider having them educated by someone who is. That way you'll be able to tutor children yourself in future and be guaranteed to get a diplomacy trait when they turn 16.
When a vassal declared an independece war, all your other vassals like you less and become more likely to declare their own independence wars. If you have a lot of vassals, this could lead to a neverending series of independence wars. Use above suggested methods to prevent this e.g. imprison every rebel until the revolts stop, strip rebels of their titles and replace them with loyal Normans (preferably with the Content trait, as they like you much more and are very unlikely to ever rebel). These rebellions can actually be seen as an opportunity to organise land in England according to your own plan. I like to ensure that none of my vassals hold enough land to be a threat - at the 1066 start as William there's a northern Duke with far too much power you should definitely remove.
When a vassal declared an independece war, all your other vassals like you less and become more likely to declare their own independence wars. If you have a lot of vassals, this could lead to a neverending series of independence wars. Use above suggested methods to prevent this e.g. imprison every rebel until the revolts stop, strip rebels of their titles and replace them with loyal Normans (preferably with the Content trait, as they like you much more and are very unlikely to ever rebel). These rebellions can actually be seen as an opportunity to organise land in England according to your own plan. I like to ensure that none of my vassals hold enough land to be a threat - at the 1066 start as William there's a northern Duke with far too much power you should definitely remove.