They will continue to lay all through the winter, though not as frequently as when the weather is warmer. So long as they sit the eggs and they are fertile, they should hatch.
The way to catch the birds is to go out after dusk when they are both in the cote - they won't fly in the dark so you can just lift them out. If you don't have a shed, you might consider keeping them in a large rabbit cage.
Another thought, if you used a homing cage or net when you first got them, you could use that to contain them in the dovecote.