The War Campaign in BfA will send you to the Alliance zones, but it doesn’t take the place of leveling up in Zuldazar. Head down to the docks in Dazar’alor to get started on that.
You’ll need to get to level 50 and be friendly with all three Horde factions in Zuldazar: Voldunai, Zandalari Empire, and Talanji’s Expedition.
You will eventually get quests to go to both islands.
Kul Tiran is probably the easier one to get.
Not really, no. There might be some PVP achievements that require it, but if you don’t care to PVP then just leave it off & forget about it.