The Aetherytes are not natural phenomenon, they are man-made contraptions. So as far as aetherytes and teleports in the sky/flying areas, I kinda think that there won't be any aetherytes since that would require us Eorzeans having been there before and therefor having already placed aetherytes in various locations.
It's unlikely (actually I think it's already been stated) that the flying mounts will work in the old areas and will only work in certain new areas (the sky zones). Making them (flying mounts) the only means of travel in the sky zones will make them necessary so that they are always being used.
From what we have seen so far the sky/flying zones seem to be rather large and made up of various flying/floating masses of land. I can easily imagine the only means of travel between them being your flying mount or the npc paid flying black chocobo that runs a predetermined course. So far, even the best made Airships have not been able to venture extremely high in the skies and have been unable to basically reach these new areas, though this could possibly change. If you've completed the Ixali beast tribe storyline then you know of a certain young heir who hasn't given up on his dream to make the best airship. That said, it might be possible he succeeds, with help from his Ixali pals, and further tweeks the progress that was made during the beast tribe questline. Hasn't happened yet, but who knows? The groundwork for the lore has been laid.
We might end up with perhaps 1 area that will have an aetheryte thoughout the whole sky zone, or each sky zone (if there ends up being more than one). The zones themselves have promised to be larger than anything we've seen so far so that part has me happy. I hope for some fun flying and exploration with perhaps some danger abound as well (so you better be careful where you land!).
Just to be clear, we won't have flying mounts in the old areas. Maybe you missed that memo. It's the new stuff where they will be used and relevant.