Not what you asked ) Most 10yo-12yo boys I know would LOVE $10 a week to walk a dog. They won't be 100%, since they're 10, and do get sick / have familiar vacations, etc... But if you counted on it as an EASEMENT, instead of "you're a business who needs to provide 5 or 7 365 perfect Attendence... You might find exactly what you need, for an affordable price.
1) Sleeping bag & wagon. Just toss the bag in the dryer when you get home.
2) Warm clothes and umbrella & rain boots with wooly socks. The walking & warm clothes will keep him as warm as sitting bundled.
3) 40 and raining is hoodie weather up here. Its not really that cold... So it affects kids a lot less than we adults who are acclimated to certain temps (kind of like if you read someone saying its wicked hot out -75 to 80- how do I keep my kids from overheating?) For adults, we acclimate to temperatures. I wore Woolley socks and snow clothes at 70 degrees when living in 110-120 degrees, and couldn't stop shaking and shivering / went and sat by a fire in a sleeping bag at 50 degrees. Meanwhile MOST people are in shorts in 70 degree weather, and northerners are in shorts when its 50 out.
I worked on a farm @ 4am breathing frost in 20 degree weather and stamping around to feel my feet. IMHO that's "character building". Uncomfy, but no harm comes from it. Just need stout boots, hat, and gloves. This is an act of love. And lesson in responsibility. When "my" horse had colic, I was out at 3am in a sleeping bag poncho, night shirt, and barn boots walking circles for 2 hours.
I believe, quite firmly, that learning to do things I disliked, for someone I love (4 feet or 2)... Has stood me in good stead. Especially as a mom, but also as an adult in general.