Just to be contentious:
Why nobody has the armor:
A) The armor probably wouldn't fit anyone else. Even if it was special (and surely Balor wouldn't be wearing just run-of-the-mill plate), no one else could use it.
B) Alric wouldn't want it because it was tainted.
C) Soulblighter wouldn't want it because it would hide his manly pecs. (Or slow down his kung fu, or something)
Why they would have done something with the body:
A) Well, they always have before. We're told what they did to most of the past Levellers--in excruciating detail. They're just gonna leave Balor sitting there?
B) The obvious counter-argument--they went to a lot of trouble getting rid of his head. The body's meaningless without it. Well, in the past they didn't just drag Moagim Reborn's HANDS to the edge of the earth, or burn what'sisname's HAT. They were very thorough--they took care of the whole package--and Balor should be no exception.