I really feel like some are being a bit over-dramatic?? Your pup is not going to die from a tiny walk in the rain. But I guess it depends on your definition of 'walk'... I mean if you're talking full fledge like 1 mile walk, then no, but to go out and potty and do business? Sure.
Here was Jackson, at probably no more than 4 months... having a blast in freezing snow (he was not shivering, nor unhappy).... it's not like I left him out there all day, to freeze, or forced him to be out there, but when you are properly potty training a dog, you do what ya gotta do. I did not want my dog to learn to go anywhere inside (though I admit for my own sake in certain weather, it'd be nice if he would go on the porch or something
) but he's always refused pads and the like anyway... even when I would try to get him to go anywhere considered 'inside' (like a garage) he wouldn't do it.
I brought Jackson home at 9 weeks old and immediately he was allowed in our own yard. Granted, we live in the 'country', and not in neighborhood where lots of other dogs have roamed, but he had 1 or 2 sets of vaccines, vet was totally fine with it, I had to start outdoor potty training, and I would've went bonkers if I had left him in the house 3 weeks after last set of shots. He had already explored so much at that point... grass, mud, rain, snow, etc. Just be smart and don't do dog parks, pet stores, and always keep them off the ground in the vets. I am always flabbergasted to find out people actually leave their dogs indoors for this long, you miss such a window of socialization opportunity.
I use a jacket in certain situations, moreso for my own sake, lol. Jax only gets cold occasionally. I think it all depends on the dog. My dads JRT mix (who has more fur, and you would think would be hardier) gets VERY cold very easily. My friends yorkies, one is always cold, the other is always fine (both under 6lbs).
So just be the judge of your own dog... know their limits, be smart about it, and have common sense.