It has been reported his limping gait was not new. We don't know what happened between 6 and 6:40; it is quite possible he walked a fair amount of distance in shoes not made for walking. Sore foot? Blister? Circulatory problems? It is quite possible he was carrying his left shoe for a very valid reason.
I have watched the raw AP video several times. How anyone could mistake him for a homeless man is beyond me.
I doubt most homeless men wear suits.
His pant leg and some of the sleeve of the suit were mentioned as dirty by people who saw him and as I saw in the video. It looks to me like he had been on the ground at some point. I don’t think the shoe tells us anything really.