You can search amazon for "cycling sandals." There are multiple brands in the space, and they generally include screws for mounting bindings on the bottom of the shoe. They are not cheap, however.
I had a pair of KEEN "Commuter II" shoes for about 5 years, and loved them to death. Naturally, they got discontinued. (I swear KEEN is doing everything possible to go out of business.) They are up to the "Commuter 4" version at this point. Shimano also makes at least one pair, as do various Chinese knock-off companies.
In general, when I was commuting I had a backpack that carried my laptop and rain gear. The rain gear included a helmet cover, backpack cover, jacket & pants, and shoe covers (as mentioned by various other repliers).
For snow commuting I wore a "space suit" made from Carhartt/Walls insulated overalls and a snow parka, plus some rubber overshoes with spikes for snow/ice and a facemask, either a beardski or a neoprene mask. (All this because I was working at an investment bank, and they kept the office temp at 80 because secretaries, so thermal underwear was a non-starter- my gear had to be "external" so I could put it on/off easily.)
NOTE: All the links I have provided are for Amazon US. I don't know where you are, so YMMV. Also, these are the "first found" links, not the actual products I own (except a few brand names) so I don't recommend them or discommend them, I'm just showing you examples.
NOTE 2: Be careful with the neoprene masks. I was commuting at 0530, and I'm pretty sure I almost got shot by the first cop that pulled up beside me in the dark and looked over at me when I started wearing my "horrorshow" mask...