Was there a social stigma for a knight or lord using certain types of weapons? For instance, was a sword considered knightly, but an axe or spear undignified? Or did no one care as long as the other guy was kaput?

Well, the sword was a knightly weapon, but knights were “cross-trained” to use axes, maces, etc. etc. So it’s more that it would be slightly weird if a knight not on the battlefield was walking around with one of those but not a sword at their hip, but there wouldn’t be a stigma.