As you can see, each rank you progress, you get a ship slot. However, you only have one more free ship that you can purchase at Rear Admiral Lower Half, even though you get a ship slot at RA upper half. As a silver player, you also are able to get a free ship slot when you play the vault in order to obtain a shuttle. If you want more after Vice Admiral, you have to pay zen for two more.
So if I was to purchase an Aeon (sp?) shuttle, that would fill the free slot, and then would I get another slot for the Vault mission, or would it just automatically use that shuttle and not give me a new one?
You'll still get the free slot when you do the vault if you have another shuttle (back when the mission came out, I and many people had Danube runabouts already - they were in since launch even if they were useless). The Type-8 shuttle you're given takes up the new slot, though you can immediately discharge it and use a better shuttle.