Hi, Nanashi here.
Family names are updated after every maintenance (normally Tue every week) so after today's (10th of May) maintenance it should display what you've set up there.
Arena is on the menu, left side - should be the first icon if you have it unlocked. You can check "Open Requirement" which you can see also on the menu, left side, bottom-most button there to see which ones you have already unlocked and will unlock after defeating certain bosses.
For the tailor's quest, I'm not quite sure about this one as when I did it, I didn't encounter any problems or it has been too long and forgotten about it, sorry :P
Should you have further questions, feel free to send me a dm on discord Nanashi#2050 and I'll try to answer them :)
For you to be able to access Encarotia's Side Quest, you will be needing to have this title. As far as I know, you can obtain the title if you have or wear an Epic(Yellow Grade) Outfit. You can get one either through events, or by fusing two Unique Outfits. This is the Patch Notes when that update for crafting Mystical Outfits was released.
You might wanna join the game's Official Discord Server to get to know more stuff from the veteran Adventurers, and be updated for some community events wherein you may get some additional rewards.