The good thing is, no helmets, show quality or not, should not be able to feed their babies. Even the one in the picture has a beak just long enough to handle it on their own
Which is always good news because messing with foster parents is a pain (especially if you don't already have birds suitable for this).
Helmets were my first pigeons, in a way. My dad got some for me when I was 4 at an auction. Needless to say, I can barely remember them, and I definitely wasn't tall enough to reach up and help feed/water them. So most of my days were spent doing that for the chickens
We let them out a few times but the hawks really hit us all and before long, they were all gone.
I think they're cute little birds and really love the crested ones. They should be fine in the winter outside as long as 3 sides of the cage are covered, or at least partially so that wherever they perch to sleep is out of cold drafts. If they have been raised outside so far, then they will handle it, as they've adjusted to the temperature change.
As far as what to feed them, whatever you feed your chickens is fine. Small beaked birds do have some issues with the larger grain though. Scratch, wild bird seed, and/or mash/crumbles/pellets are fine. Large corn may be hard for them to swallow though. So really the wild bird seed is great because of the milo and millet.
The most important thing here, if you plan to breed especially, is calcium. Some grit has crushed oyster shell in it already, but if not, you need to mix some in. The crushed granite is still important though because it is the digestion aid (oystershell doesn't help much with that because it's easily broken down).
Is it safe for your chickens to have these little guys near them? Yep! Although as with any new birds you get, it's always best to quaratine them first. You can put the pigeons and doves through a round of a broad antibiotic like Terramycin, or maybe just kill a lot of birds with one stone and use Sulmet, since it'll take care of coccidiosis too.
I've never personally had any of my chickens get sick from my pigeons or vise versa. The two big things that are easily transfered between poultry and pigeons is worms and coccidiosis. Pox could also be an issue if you get a bird with it. Whether that bird is a chicken or a pigeon, isolate it until it's gone. I don't know that chickens can get pigeon pox, but I'm fairly sure both can get fowl pox (really, both of these as well as canary pox are all just strains of the same thing
). Anyways...pox isn't very common anyways, at least not in well-kept coops and lofts. Just very contagious from bird to bird.
I have kept pigeons in an aviary with chickens, quail, and pheasant before. Everyone got along just fine, as long as there was plenty of room, perches, and food/water. If you (or whoever ends up with them in the end) are planning on doing this, it will help to have a feeder off the ground for the pigeons so the chickens don't get all huffy when it comes time to eat
If those 'doves' you're getting are actually ringneck doves, I'd take them!
But I don't think I need anymore pigeons right now, LOL.