I don't know either why that represent offsprings, especially sons. Someone told me it's because pronunciation of jujubes and son in Chinese is very similar, but not sure if it's true. ;)

I don't think it's a skirt they're suspending to catch chestnuts. It's called 활옷 and, well.. You could say it's a kind of coat or something, because the bride wears 저고리 and skirt inside. Then she wears 활옷 over the skirt and 저고리.