What do the various hangers-on and other minor members of the court who seem to have no actual function do with their days? Are they doing work in the background or simply lounging around the Red Keep whenever court is not in session?

Good question! 

So I’ve talked a bit about how you become a courtier, a bit about various offices, a bit about how you would get paid (or not)…But let’s say we’re talking about someone who isn’t one of these people, who is literally just a hanger-on: what do they do with their days? 

A big part of what they’re there for is access. Since royal politics is far more organized around proximity to the person of the monarch and, further out from there, proximity to people who are proximate to the person of the monarch, just being around the court means that you can present petitions, ask for favors or money (you’d be amazed at how good nobility were at mooching), and (more importantly) get money from less important people for doing it for them. (There’s not a huge difference between being a courtier without office and being a lobbyist.) But mostly, you’re hanging around waiting to be noticed and given your big break, just like show business.

So while you wait, a big part of your life is entertainment. Remember, these hangers-on are nobility; by definition, they don’t work for a living and would be horribly offended if you suggested that they should. So they have money from the family’s estates, and they spend their lives in the pursuit of pleasure – and to do that, you need a big enough concentration of highborn folks that you can do social events. So there’s hunting, feasting and drinking (and other recreational substances), dancing, having affairs, GAMBLING (for reckless gamblers, you really can’t beat that combination of cultural disdain for money and aristocratic competitiveness), amateur and professional arts, and other organized activities, and some of them are socialized as male and some as female and some as mixed (because courts are also marriage markets, because one of the ways that people who don’t work make their way in life is by marrying well). And as smart people like @goodqueenaly and @nobodysuspectsthebutterfly have written about, these entertainments had symbolic political functions, which is why people paid for them to happen. 

And another big part? Gossip, rumor, and public opinion. In the absence of a news media, it’s handy to have a big group of well-connected people who have nothing to do but talk about what’s going on. You’d better bet that there are ambassadors who hang out with or pay courtiers-without-office for the latest scuttlebut about what the king and his family and his officials are up to, so there’s real value here. Likewise, if you’re the king, the royal family, and the government, without any way to assess public opinion, the court is the only sounding board you have – and it’s a sounding board that is connected to the broader political class, because all of these hangers-on will talk to their relatives and peers back home – so it matters if a proclamation or decree or policy is very unpopular at court. And since no one’s getting elected, the standing of any official is their popularity with the court, so rumor could “make or mar.” 

So that’s what the idle rich did with their lives.