I got my first mount from Lorenzo Murray on one character. Now I am levelling a second character and it appears Lorenzo will not give me a second mount? Does he only give out one and the other characters you level share it? If that is true my first character is still sitting on the mount. Do I have to check it in to a stable in order to fetch it for my second character? The Stable screen is so confusing: it says at the top 'No mounts currently taken out' and at the bottom 'Mounts taken out 1'. Why the contradiction? Appreciate any help as I find even the simple stuff in this game confusing.

The free mount is once per family.  Mounts and boats that are out with characters and not checked in to the stable or wharf will not be accessible to your other characters. 

The characters with mounts/boats checked out will have a horse head symbol or an anchor by their name in the character selection list.

The contradiction is likely due to the character you are on at the time having not checked out any mounts, but the alt that is sitting on your horse last checked that mount out of the Velia stable.