Wrong, NPC damage is persistent, many players (including myself) have tested this. There are some bugs though.
No, it is not. I'm sorry but I've been interdicted more times than I can count and NPC damage is not tracked from prior encounters. The NPCs are not persistent in any way, they simply insta-spawn when they need to be created. This is how they can show up behind you suddenly during supercruise, how they can "follow" you on 30 ly jumps (they are simply re-spawning) and even how an NPC with the same name can reappear with a different combat rank. These issues are not "bugs" they are limitations of the P2P architecture that creates new NPC spawns when your CMDR enters a new instance, which means every time you jump, drop out of supercruise, drop into a USS and so on.
There are dedicated servers on different layers, the first layer handles the BGS stuff, the next layer handles matchmaking etc and the last layer is your P2P connection which handles NPC, USS and player interaction. Your actions are still reported to the server rather than the player hosting the instance.
Yes, there is synchronization between the server and the client re. BSG, system state and so on but the servers FD uses do not track or store the detailed information about any given NPC or any given mission. That is the entire problem with the lack of persistent NPCs, the servers are simply not set up to handle that at all. They give your client the information it needs to do most of the work on its own and FD would need a massive investment in central dedicated servers to track all of this persistently.
NPCs are generated in your instance, mission generation is not. You can log in and log off and the bulletin board will still be the same. It only changes if you are connecting to a different server (which often happens during mode switching).
Yes, some mission information is retained, for example, if you have a mission in your inventory it will stay there despite any number of mode switches and the mission boards only refresh at certain intervals unless you switch modes. That does not mean that it is not generated locally, for example, when driving the SRV or flying around in supercruise you only see "updates" for the rocks or USSs at certain intervals but the rock or USS is still generated by the client and they only exist as long as you are in that particular instance. For missions the process is slightly broader than this as the information is retained between stations, much like inventory in the "buy back" option staying around for 10 min or so, but it is still tied to your CMDR and will disappear once the server tick causes it to refresh. That is also why we have problems with the "0 min 0 sec" mission timers, they are triggered to be generated on certain conditions such as getting close to the delivery station or system but are not tracked persistently in a way that would make sense. They are just flagged for the client to generate at a certain interval even if it makes no sense due to being impossible to complete the mission in that timeframe.