she will be fine in your garden so long as no other dogs can gain access to it. A pup of that age can pick up all kinds of nasty bugs because they are too young to have had all of their vaccinations.
On the cold fromt, it sounds silly, but you can get doggy slippers to protect their feet. I can't see you needing them as long as you can get puppy out, wait for her to have a wee and then call her to you making a big fuss and ger her back in for her feet dried.