Domus Flau is the Arena of Crocus. Travelling to here counts as Crocus since it is practically the same city. However when you are here, you can post in Crocus and visa versa. 

Domus Flau is for you to fight! Battle! Duel! You can go and demand a fight with someone, you can invite someone to have a battle with you or you can request for an NPC. It will be a surprise what kind of NPC you will get. 

You can use only battles/duels or fights for stat training. You will get a 10% discount for this. However it needs to be only a battle. After the battle buying ice cream doesn't work. Also the battle training follows the normal training rules in the matter of: only one training topic per time.