Men can wear pajama bottoms, or a smoking jacket and satin pants (a la Hugh Heffnerr), but most men wear boxer shorts or boxer briefs, ideally in some fun pattern (tuxedo) or color. A few brave men will wear a thong. No tightie whities, please! Too utilitarian for TTR!
I wore the robe after a shower. My husband likes the A/C on very cold so a robe is nice to use instead of wrapping a towel. But not essential. I would not worry about using a robe for lingerie night, too hot for that heavy robe outside the room! No one will mind seeing you walk there either