I 100% current build.
1. You can't do anything there right now.
2. No, letting Velvetia possess you doesn't do anything right now.
3. The most impact I've seen with the fox shrine is having speed drastically increased. There is a bar with the body influence levels, if you max it, it increases the speed characters have when they are possessed by the same spirit.
Take Galiene for example. Base Galiene has a speed of 129 at 90, but possessed Galiene has a max speed of 221 with a full bar.
Take note that you do not have to battle to increase the body influence levels. You can just fast forward time to increase it.
Hope that answers your questions.