dogs can be trained to use litter pans and there are actually special dog pans and litter. why is this vet telling you all this stuff when there is no real problem is what I want to know. I would insist my dog potty outside unless the snow is snow is so deep and horrid that you can't stomp out a small area to go otuside the door. unless the conditions are so cold that human skin flash freezes i don't get that it would be to cold for a little doxie esp if he has a coat. You don't need to do a 30 minute walk, just a minute or two. doxies are tough little dogs bread to hunt and though you couldn't leave him out in cold weather a quick potty trip is not going to hurt him.
i would be more worried about teaching him to go inside and then expecting him not to generalize and think potty in the house anywhere is ok. esp for a smaller dog breed it is less confusing in the sometimes more difficult potty training to get reliable results. now if you had a garage where u could keep a litter box that might be different but I would train outside like any other dog and not go through all the extra weird stuff this vet is suggesting. it sounds like more levels of complexity for problems that don't exist now and may never exist.
now if you are gone all day and can't give him a break there does need to be some provision for potty. I would prefer to crate train, come home for 1-2 potty breaks a day, or go to doggie day care. what a journey here??
litter for dogs is different than cat litter. most people use a pellet made from compressed wood fiber and recycled newspaper. these pellets should NOT be eaten because they do expand when wet and could cause intestinal blockage which can be fatal. cat litter is the same and a little more dangerous because it is designed to clump when wet and makes even more solid blockages. dog litter also has little smell inhibition so i wouldnt really want it in my house unless i wanted it to smell all the time. I built an outdoor potty box on our pation for the dogs to go in the winter/snow. I use shredded hardwood bark mulch but our puppy still chews on it. She was eating the pellets so I quit using them. We don't have a feced yard but have really large pation so it's a perfect spot and easy to clean.
I really wouldn't train the dog to go inside, if I did I would probably use a pee pad that is is either in/under a wood frame so it can't be eaten or taped really well to the floor. The litter will be everywhere and it turns to pee soaked sawdust when it is wet. .