1. You don't have to do any.
2. You don't get much of the important factions from killing.
3. You only have to do the ones on the charts with solid borders if you want to raid... but you need the faction, which is most easily obtained by (drumroll) doing the other missions.
4. I have (it seems like) 20 open quests in my quest window, when I've never had more than four before last month. Ashengate and Frostcrypt seem to be relatively unrelated to each other, from what I've read, they are just alternate XP/Raid zones, and you are expected to be Ally with both (and will need to be in order to raid both). It's not like CoV/Kael, where it's either/or.