I noticed that it did not seem possible to complete the Ebony Blade quest after the Battle of Whiterun was completed and a new Jarl was installed. Are there any others like this?

I picked up the quest from the Bannered Mare at some point after Jarl Balgruuf indicated he would be leaving Dragonsreach within a few hours.

Hulda mentioned the rumor about the "Jarl's children" and I was wondering if perhaps she was referring to the new Jarl at this point. When I returned to Dragonsreach, Balgruuf was nowhere to be found; however, he apparently left his children behind in the basemant! (Can you really blame him? )

None of the NPCs at Dragonsreach seemed to be able to help me proceed to the next stage.

I also noted that despite being Thane and having Lydia as my follower, Vignar Greymane mentioned he would give me a new housecarl. If I pursue this, will this remove Lydia from service?

After the battle the Jarl can be located in the basemant of the Blue Palace, or just steal the key you need from Farengar.

Lydia won't be removed from service, she'll remain your housecarl.