I finished the Companions quest line a long while ago. And at the time, I only carried one Glenmoril Witch head with me because I was too lazy to kill more. So I released only Kodlak. A while later, I experimented with the beast form and decided that it was pretty useless and prevented me from being well rested. So now I went back to the coven and collected 3 more heads. But I can't reactivate the flame in Ysgramor's Tomb.

The UESP states that it is possible to get a quest from Farkas or Vilkas about curing their Lycanthropy, too. And if I bring them to the Flame, I could reactivate it and cure both my companion and myself (if I'm carrying at least 2 heads). But Vilkas only offers other radiant quests and Farkas doesn't offer anything. What am I missing here?

If you have the witches' heads in your inventory, the brothers should offer the purity quest, but they may not do it at once. Especially Vilkas' quest is bugged, so it's best to help Farkas first, and then Vilkas immediately after. Since Farkas doesn't offer any quests at all, my advice is to start by doing a radiant quest for Vilkas, then save before checking with Farkas. Hopefully he'll offer a radiant quest, and if do, keep doing radiant quests for him until he offers the purity quest. Make sure to save before reporting to him, in case it bugs out. After you helped him, go directly to Vilkas and do radiants for him until he offers Purity, and THEN you'll be able to cure yourself.

If Farkas still isn't giving any quests - then I can't help you, but maybe somebody else can.

Sometimes it takes a while for the right quest to pop up. I generally do Aela's totem quests while waiting for these to come up, though I don't believe the totem quests are required.

I can confirm that they are NOT required.

Yeah, ditto on that. I've done the Purity quests for both twins without ever touching the Totems of Hircine, because on at least one playthrough I cured myself when I cured Kodlak so the Totem quests were never even available. Still was able to cure both twins. Have also waited until curing the second twin before curing myself.

But getting Purity from Vilkas first is likely to cause problems and so it's to be avoided. Take quests from Farkas first until he gives it to you instead. If you need to get radiant quests from Vilkas in the meantime, just store the heads somewhere when you talk to him. Neither one will give you the Purity quest if you don't have a witch head in your inventory when you ask them for work. I've delayed doing it by not carrying the heads around with me until I want that option.

I think I figured out why Farkas wasn't giving me anything. There was a quest buried in my journal somewhere that I accepted a very long time ago but never bothered with that he must have given me. I finished that and now he has offered the purity quest. But even after curing him, I can't activate the flame to cure myself afterwards. Another strange thing is that none of my 3 witch heads have been removed when I cured Farkas.

You must cure both brothers. Go straight to Vilkas and get the quest from him. Make sure to save first in case he glitches (the bug is that he stops following you and disappears, but it should work if you cured Farkas).

The witches' heads in your inventory is another known bug. I was able to simply drop them afterwards and then they disapperad, but I've heard people who are stuck with them. Fortunately, they count as quest items and are weightless.