I was wondering why is it that if I make a purchase of an item, in this case the featured Enterprise ship, and I remove the item from my character I cannot access it again on that character? I am pretty sure buying that item should give me the right to do whatever I want, and re-download it. Please someone explain this to me.

Check the "Special Unlocks" tab of the C-Store. It should be claimable on all of your (Fed) characters, as long as you don't already have it in your inventory or ship list.

If you delete it, you can re-claim it again on the same character. This is done all the time.

There is one caveat to this... You must always have at least one ship at all times. If you claim the Connie as a starting Lt. and discard your original light cruiser, you will not be permitted to discharge your only remaining ship or claim a 'duplicate' Connie...