What’s the difference between a “Socialize” and a “Socialize!” activity

I believe that core quality refers to the item itself - i.e., a superior-crafted door, and total quality takes into account the masterfully encrusted with blue diamonds and other miscellanea.

I've been using "Core Quality" for a furniture stockpile of masterwork doors (admittedly, in 31.16) and only masterwork doors are going to that pile.

As the Wiki says:

Nobles are snotty good-for-nothing parasites

Jokes aside, noble dwarf are quite demanding, but they are useful because they unlock the possibility to appoint other nobles.

Your baroness, as you can see in the table in the link, will let you appoint Champions and Dungeon Masters (and Tax Collectors in the future)