It does happen. Tywin Lannister took Eddard's sword "Ice", melted it and forged "Oathkeeper" and "Widow's Wail". Obviously he did not sell it - valyrian steel is nearly priceless and on top of that he is a Lannister, therefore loaded.

In normal combat that would not happen very often - the swords used by the common soldiers appear fairly standard and thus of little interest to either lords or commoners.

I suspect that the bodies of any lords killed in battle are claimed by the winning lord. I don't have any references at this time, I'll try to find some later tonight.