I'll cover these as best I can even though I'm only just approaching ten hours in. I think I have most of this down by now, through either experience or research.
1. When a unit dies permanently it opens up their party slot and you can return to the castle (or other locations, once you unlock more classes) to either recruit another member of the same class or a member of a different class. As units within a given class/gender combo die they'll progress down a list of characters (each unit has a fixed name, stats, and weapon preference) until it's exhausted in a random order, then start over. Everybody's somewhat different, even within a class.
2. There are two separate successor picking situations.
When you complete a certain number (varied, lightly randomized) of plotlines, a new era will be ushered in following a several decade jump forward in time. This locks in the techs you've learned for weapons and allows you to learn them on your future characters at no cost in the training hall. Your objective for each era beyond completing area plotlines is to spark as many skills as possible and raise your proficiency levels in various elements or chosen weapons as high as you can, as these increase globally. As you get stronger, your future parties' starting point grows stronger.
If your emperor dies, you simply pick a new successor within the same era from a list of four of the numerous available units. Nothing previously learned that era is locked in for non-emperor characters in the case of a total wipe because the era hasn't changed, so you start from the same rough point you did at the beginning of the era. The only reason you might want to kill an emperor deliberately during general play is to change to a different class, which can be used to unlock new formations in the training hall in the castle for some classes.
The game is balanced assuming you're going to have some deaths--and even total party wipes--but enemies get stronger with the number of battles you've entered so while the game is pretty lenient about it, you probably shouldn't run around having massacres visited upon you three or four times an hour for multiple generations in a row any more than you should spend hours grinding fights against less formidable opponents.
3. It goes into storage at the castle, same happens when the era changes over.
4. Different weapons have different niches in their skillset and different damage types, yes. The game goes out of its way to hide as much of this as possible, but RPG logic generally applies. Use blunt weapons against skeletons, projectiles against avians and flying insectoids, and so on. A character will tell you what weapon he or she is good with when you recruit him or her, so try to have a decent spread of coverage when you set up your teams.
5. You are still in the tutorial for a while yet, but classes already exist and all of your characters have them. You can recruit different ones when they die or once you switch over to the next ruler.