What dog is the cheapest?
There is no cheapest small dog. Dogs are usually priced on their characteristics, breed, and colour variation. They can also be priced accordingly for what they could make, like a very active one being priced higher than his lazy brother for example. The cheapest dogs are usually ones from rescue centres, all you need to pay is a small donation, but the dogs are more picky and generally harder to train.
If you are looking for the "cheapest" dog, you likely should not adopt one as any dog can get expensive- vet bills, food, etc. Many dogs end up abandoned because people don't realize this. Unless you can make the commitment to treat the dog (or cat) as part if the family, meaning treating it well and loving it and not abandoning it for ANY reason, you probably should think of another idea.
It depends on the weight of the dog and the Veterinarian. Call around to different Vet offices and have the weight of your dog handy and see who is the cheapest. I would also inquire of your Local Animal Shelter when they will have spay or neuter certificates out to help you with the cost of neutering your dog.