Becoming Thane is a quest you can do for any Jarl in their Hold. In most cases it has to do with a favor for the Jarl plus favor quests for various people you meet. I don’t think a dead Housecarl gets replaced. I’ve never had one die unless I was using them as follower.
You might want to ask this in the spoilers section so that you can get more specific answers. There are some differences between Holds in how you get the Thane quests. For Hearthfire houses you would be looking at Falkreath Hold, Hjaalmarch (the Jarl is in Morthal), or the Pale (where the Jarl is in Dawnstar). You have to be at least level 22 to get the quest from Jarl Skald in Dawnstar.