When mounted and armed with a weapon which do both thrust and swing damage, I only seem to be able to make stabbing motions. If mounted and armed with a weapon which can only do swing damage, I make swing attacks. I suspect it's a setting I changed when new to the game, but which one? Or is the problem something else?

I don't know how new to the game you are, so... Have you tried enabling attack direction arrows in the options for better visibility and then attacking from direction other than from below?

I can make other attacks with the same weapons while dismounted, Helpo, so it doesn't seem to be how I'm moving the mouse. And I just tried it again while mounted, but only stabbing motions resulted. Thank you for the suggestion, though.

Now I am at the castle where the lord is telling me to see his daughter in her chambers. But there's no daughter and no chambers. Any help?

Ugh. I see others have had the same problem, so here's the solution. She's right there in the same frikken room, but only if you go outside and re-enter by choosing the new option to visit a lady.

Does this apply to all weapons (including swords)?

Polearms are normally set to work this way So, for example, if you have a wooden staff and no shield (both hands free), you'll be able to swing it in melee but not while mounted. While mounted, it can only be thrust and should ideally be used to deliver couched lance damage, because that's the most efficient attack while on horseback (in Warband).

OTOH, a sword should work the same in melee and on horseback (both swings and thrusts). If swords, one handed axes etc. don't do swings when you're on horseback, check the setting: "Control Attack direction".

1. The hafted blade is a unique polemarm – AFAIR the only one that can be swung on horseback.
2. All one handed weapons should be swing-able.
3. Some two-handed weapons (that aren’t classified as polearms) can be swung (e.g. the maul, the voulge). I generally don’t use them, so I can’t specifically list the ones that can/can’t be swung.
4. I don’t remember what the war axe is like and can’t test it on my Vanilla Warband savegame right now. Just to be sure, I’d check if the axe has an alternate attack mode – by default the key for switching modes is X. If pressing X doesn’t allow you to swing it on horseback, I would assume it can’t be swung.

You're correct about point 4, the long axe and long great axe are dual mode (polearm and 2H) so you hit X to toggle modes. Excellent weapon BTW; the long great axe as 2H does 54 damage with 127 reach IIRC.

Thanks for confirming.
I generally prefer one handed weapons + shields, with ranged weapons (crossbows) as a second option. Doesn’t pack quite as much of a punch in one go, but I like the versatility. So my memory of which two handers do what is spotty at best.