My Chocobo is now in Rank 10 and he doesn't gains more exp. What I need to do now? I have a Thavnairian Onion. I think it unlocks the way to the next rank (in my case, the Rank 11), right?

But, one thing... I read that Free Companies and Housing are related with the Chocobo training (well, I mean chocobo stables).

It's obligatory the chocobo stable? (I'm not in a FC and I don't have a house). What happens if I give the Onion to my Chocobo right now without the help of a stable? Will it unlock the Rank 11 anyway? And if that so, what's the difference between giving the Onion without a chocobo stable and with one?