I'm sure someone with more experience will reply soon, but what I know is you really can't tell until they get a little older. A female will start to lay eggs. I think, sometimes males are a little larger and will start demonstrating courting behavior.
I'm new, live in the south and keep my birds inside, so cold weather baths I'm not sure of. Someone will reply soon I'm sure.
I believe you can start to tell the sex of a pigeon once its about 6 to 8 months old.
That's when they are sexually mature and start to show signs of what sex they are.
I let my pigeons take a bath year round.
If its a really really cold windy day I wont let them bathe but if its a sunny day but not THAT cold I let them bathe early in the morning so they have most of the day to lay in the sun and warm up a bit and also to make sure they are dry WELL before the sun goes down and things get colder.
Usually if it drops below the 50's, I stop giving baths. It's not so much the cold water as it is being able to dry quickly, I would think. I have seen some wild birds take baths in some colddd weather! But either way, they seem to know when it's too cold for them to take a dip
When you can sex them depends on the bird really. Some will trick you for a long time, and others you can tell right off the bat. Going by size isn't always a good way to tell, as pigeons come in all shapes and sizes. You can start getting accurate guesses at sexual maturity, so about 5-6 months old. Usually the guys will show themselves by then, although some hens can be cocky. Also the pelvic bones usually tighten/loosen at this point to give you a better guess. And you'll get 100% accuracy once the hens lay eggs and the boys start strutting their stuff/mating with the hens
Bathing helps clean dirt off the feathers improving their insullation, so I offer baths several times a week as long as it's above freezing, early in the day, and not windy. The only one I have to watch out for is Walter, who has a preference for showering in the rain late in the day. I'll find him when closing up, dripping and shivering, so it's inside for a blow dry.