There is one: If you make him a duke, you will create a republic that you can actually play (by saving and loading as this character) if you have "The Republic" DLC. This can be quite nice if you don't like the few default ones.
Even when not playing it, I guess it should be quite wealthy thanks to trading post, ultimately a bigger boon to your income than a regular duke.
Hmm, never knew that - I like the roleplaying a dynasty thing too much to reload as a new character halfway through a game, but it's nice to know the trade post option actually has a function for non-Republic states.
I am trying to assasinate him so the second son takes his place (Harald has like 15 children) but with not luck so far.
Oh dear. You're about to experience the wonders of gavelkind succession - it will be a good idea to assassinate the other thirteen as well.
Yes, losing a high-prestige character is a blow, but a portion of their prestige is passed onto their successor. I think it works proportionately, so if your king has high prestige when he dies you should have a fairly stable time as a successor (gavelkind aside). The most important thing is to be sure you have high diplomacy and/or a chancellor and/or wife with high diplomacy. In my current game Alfred held the kingdom together (he was a very good - and long-lived - chancellor) since I had very high diplomacy once my first king died.
I suppose you did not land in the north, and march to Stamford Bridge then
Wrong Harald, if it's the 867 start - that's Harald Fairhair, the founder of Norway (who consistently founds it in the game, usually with a high degree of stability). Hardrada has no choice about Stamford Bridge - he starts there in the 1066 start. He usually survives, though, even though he'll generally lose (he did briefly become king of England in one of my 1066 games, only to be sent back to Norway by William). I always like to keep an eye on Tostig - it would be nice to play him, but he's a pretty lousy character and he's one of those characters - like Bishop Odo in 1066, or Ubbe Ragnarsson in 867 - who tends to stick around as an eternal courtier, and isn't often given territory if he's got an AI liege.
I married off my first son & heir to the HRE's emperor's daughter.. She died a couple years later giving birth to a baby boy, who is now 5. It sounds like I should keep this guy around until he's maybe useful and not put him in charge of any vassals... right? Could he be useful? Could I assassinate the Emperor's two sons and then all the other male grandchildren he may hav
Yes, you can do that, and it can be a good idea - I put a relative in charge of Byzantium that way. But there are two things to be aware of if you're thinking of inheriting - one is that you can't ever have an emperor as a vassal, since it's the highest rank in the game. This means you'll never inherit the HRE through a vassal (such as a relative) even though you may inherit claims to it; the only way to inherit an empire is if the same character is the emperor of one and the heir to the other.
Secondly, you don't inherit the claims of grandchildren, only of children - so your son and heir will have a claim on his son's behalf that will let him go to war against anyone you don't assassinate to claim the empire, but you won't.
Also, if the reason you want to do this is to claim the HRE, the best thing to do is to kill off your son's other children and perhaps make sure he doesn't have any more (e.g. by killing or divorcing his wife and not remarrying, or marrying someone too old to be fertile - over 40 or so). This way, when your son dies his only heir will be the one who runs the HRE, and you will then play as that character and have a claim on the territory you started with (if you don't inherit it).
To demonstrate the above, say you're King of England and the heir to the Kingdom of England is the heir to the HRE, and the heir becomes Emperor before you die. Then you die, and as your heir the Emperor now inherits the Kingdom of England. If instead the Emperor's son was the heir to the Kingdom of England, because a kingdom is a lower rank than an empire, the Emperor would still inherit England; however he wouldn't inherit the Empire of Britannia, if that has been formed, unless he personally was the heir - if his son was, then the son would become independent as soon as he inherited.
Be careful though, since having only one surviving heir until late in your likely lifetime is very risky; if they die you can lose everything, and if they lose their own claim or position before you take command of that character, you can end up with a useless duke who may have a lot of claims, but probably doesn't have the territory, demesne limit or army to press them successfully (I'm still cultivating my deposed relative in Byzantium, who has reams of claims - actually his son is the relative rather than him - but I'd need to go to war with the empire to claim it).
If your character starts with stewardship below 8, he can set himself an ambition to "Improve Stewardship" (I think ambitions might be expansion/DLC content, I'm not sure - you should know if you've had a notification that you need to set an ambition). This will trigger occasional events that give you options to increase your stewardship or to gain traits that have that effect. Otherwise, random events can sometimes offer traits that do the same thing. If you have Sons of Abraham, you can choose to go on a pilgrimage: this is basically a way of forcing the game to give you a sequence of random events that can result in you getting good (or bad) traits. There's no guarantee that these will include events that improve your stewardship, but they may do.
I haven't tried it, since I always "spend lavishly", but the holding feast and tournament event chains offer options for spending less - it's possible that selecting these options will give you a chance of improving stewardship.
Thinking ahead to the succession, if you want your character to focus on stewardship (personally, I wouldn't advise it - hire a good steward or marry one; I personally always focus on Diplomacy to keep vassals in line, and Martial to improve my armies), give him a tutor (when he's a child of 6+) who has "Midas Touched" or, if not available, another stewardship-boosting ability (this is another good reason to marry a steward - she'll then be available at court as a tutor), since a child has a higher than average chance (but not a certainty) of getting the same "career" trait as his tutor (this is the trait that's square, and at the start of the character's trait list - it will be green for abilities that improve stewardship more than other stats).
Ideally you'll have a career trait yourself that you want your son to have, so that you can tutor him - this means that you get random events triggering that let you select the types of personality (i.e. non-career) traits he receives. The events that trigger are influenced by the traits the tutor has (so, if the tutor is honest, an event that gives a child a chance of being honest will usually trigger) - again you can't rely on getting similar traits in your children, but by choosing tutors with good traits (if you don't have them yourself), you can improve their odds of getting them.