Hi everyone,
I don’t know if the problems is only on my side but I can select a livery only once. Meaning, if I beat the demo and restart it, the said livery is no longer available from the list. So, for exemple, my S2000 car only has 1 livery available now (the only livery I never chose yet).
Bug?
So, I guess I’m the only one with this issue. 
There was a similar bug in FH2 wherein one would search for a livery, whether or not it was downloaded, and after it was viewed it could no longer be found in the storefront. If it has not been fixed then it’s a real shame.
Funny, this has always happened to me in Forza 5 as well. I love things like this (bugs, glitches) that persist for several games in a franchise. Makes you really wonder if developers pay attention or even play their own game(s).
sigh
Indeed, liveries is a big part of the Forza experience afaic.