The order is always Limsa > Gridania > Ul'dah, regardless of which city you start in. If your friends can enter Tam-Tara then they can definitely enter Sastasha. If they're saying they can't then they likely just don't want to help you.
And no, the player who initiates the conversation is irrelevant, all party members must be up to or past the relevant quest in order to enter a specific dungeon.
Thanks so much Lykain for clearing that up! My friends were still at level 10, so we didn't know that they would have the same mission progression as me after getting airship pass. Sweet, we can start grouping for dungeons next weekend.