I have declared the witch hunter to be OK in a game I DM and we freely loot other systems for things like bonds between characters. But for actual rules I don't allow any 3PP and I likely won't use such adventures or other DM material either.
Basically thus far it's only the Core books, SCAG, UA and EEPC that are allowed. I would also allow the background trait from the Haunted background in CoS.
I've used a few classes from different sites including the revised elemental monk. I enjoy the monster a day I found from reddit which has had some interesting monsters. I worked on a shaman class for while but the player decided not to use it so I left it. I always vet new things a player brings me, and I'm usually open to trying new things.
I should take a look at the Foes book, seeing how it was well received by this thread.
Like many else, I use Fifth Edition Foes (and will use Tome of Beasts, when it comes out), but would probably not allow a player to bring some random third party option to the table. As it was, until recently I didn't even allow the use of WOTC player options outside of the PHB.
Yes to whatever I find interesting. I have a host of race, subclass, monster, spell, and magic items options available for use. DMs Guild, the EnWorld Homebrew Forum, blogs, whatever I find that I like is fair game.