I have a short haired mutt who gets cold easily. How can I figure out the temperature when he needs a coat and when he doesn't need one?
You can't really. Cold is subjective. I work outside with my horses a lot and after years of going out and doing them in the cold and just pushing through, I've found that I can ignore the cold. I'll be outside working and I can feel the coldness, but I tune it out like you do a noise. On the other hand, I work with people whose teeth literally chatter at 68 degrees.
It'll be similar with your dog. It'll depend on whether he's been out in the cold and is used to it or whether he's coming from inside a warm house. It'll depend on his personality and what he's okay with. Breed plays a part as well. I know a guy with a yellow lab and his coat isn't particularly long, but he'll go diving in the river in the dead of winter. Labs have a good bit of mass and oils in their coat to keep them warm. Our keeshond has a couple of layers of coats to keep her warm. My Jack Russell, well let's just say that you can see her freckles when she's wet and she doesn't have much body mass. She fakes being cold a lot. I don't know how she learned to do it, but you can peak out the glass door at her and she'll be laying there looking around, but if she sees you, then all of a sudden she's shaking uncontrollably. It stops when she thinks you're gone.
I would personally use a mixture of common sense and any signs you see. If you have a smaller dog without a lot of hair or other protection and you feel cold outside, chances are they're going to feel pretty cold. Keep in mind that jumping outside in the cold and right back in is no the same thing as being in sustained cold. You may jump out of a hot shower and not feel it much if you go out to crank your car in 8deg weather, but if you're outside all night and it's 60deg, you'll get pretty chilly.
Look to see if your dog displays signs of being cold, including shivering, trying to curl up in a blanket or somewhere else warm, or anything else you believe is your dog showing a sign of being cold.
As for physical evidence, like with a lot of animals, dogs have large-largish ears. These are thin and have small blood vessels running through them. If you're hands are an ambient temperature and her ears feel cold to you, chances are she's cold. They will get cold before the body will, due to being further from the core and less blood flow.
I sometimes put a jacket on my JR and she'll wiggle and worm her way out of it when she warms up. So you'll just have to use common sense and observation to determine when your dog might be cold. Good luck.