I'm not sure how far you're walking the dog...but I had a baby last December and she and I walked her brother to the bus stop every single day. I wore her in a Baby Bjorn or an Ergo, put my snowpants on without zipping them (over her feet) and then wore my husband's big bulky coat. I zipped it around her--but her head was definitely out--and then put a good hat on her head. We were pretty much out on all but the absolute coldest--like 40 below or really windy--days. Even some of the -30 days I think I took her out. She just didn't like the wind. Now that she's almost 1, we're going out again, and today it's 12 degrees out right now. We're out for about 10 minutes. I don't know whether your "one" is just barely one or almost 2...if you can find a carrier and a bulky coat, I'd think that might help on the coldest days. If baby is almost 2, I guess I wouldn't worry too much. Oh, i just see--30 minutes tops. I would say you could go out to at least 10 degrees without too much worry; I would let baby guide you a little too--my baby is getting chapped cheeks, which bums me out, since with two older brothers, I'd like to get out more, but I'm not sure her skin will like it.
ETA--the rule for my kids' school--including my three year old's preschool--is that temps or windchill below zero at recess time equal inside recess. So, in our area (north of you a ways), 0 is considered too cold for kids. Now, I know plenty of kids--little kids--are out playing organized hockey, or ice fishing or sledding below that, but for some of the more careful and more conscientious groups, zero is the line. But--many of those activities last more than 30 minutes...so if your kiddos don't seem to mind, I'd say you could go colder than 0 if it seems to work for you.