If using weapon skills while mounted, is the skill level on those weapons somehow limited by the one's Horsemanship skill level? An example: a knight wants to use his sword skill (15) against an opponent while on horseback (Horsemanship 12). Does this mean the effective sword is know 12?

No. As a matter of fact, you get a bonus for being mounted against an in mounted opponent.

No.

But when using a non combat trained horse you need to roll horsemanship every round to keep it in control.
And as Headwound says: you get a +5 modifier against unmounted foes. And they get a -5 to hit, unless they have a great spear of something alike.

Btw combat trained does not mean it will attack, it is trained as to not be spooked by the combat and loud noises nearby. they still obey their rider. That is also the reason you do not have to limit the sword skill by the horsemanship. Only attack trained horses do have an attack score by themselves and can assist the rider during combat. These last horses are of course very expensive and very rare.

I don't know where I suddenly got this idea that it works as in the example I gave, maybe it's a game mechanic in another rpg I've played through the years...? Anyway, thanks for the clarification. And I knew about the reflective bonus/penalty for mounted against 'grounded' opponents as well as the Horsemanship roll for horses not combat-trained. But I do appreciate your explanations!

It feels like the rouncy is the 'main combat horse' of the Uther era, although everyone wants a charger.

Yes. everyone wants a charger. But they are even more than the ransom you must pay for a vassal knight. ;)
You will probably see them not using their Lance skill in the early years, if their damage is higher than 4d6. At least my own PKs are less on the charging side at the moment.

You will probably see them not using their Lance skill in the early years, if their damage is higher than 4d6. At least my own PKs are less on the charging side at the moment.

Yes! That's exactly what I discussed the other day with one of the players. Its serves to show, in my opinion, that the lance charge and hence jousting, had yet to reach fame among the mounted equites. It is wondrous things to come.