A sire is a male dog who is the father of a litter of puppies.
If by "Session" you mean estrus/heat; no. A female dog can not get pregnant if they are not in estrus. Many female dogs can not safely carry a litter to term if they where bred during their first or second heat, and reputable breeders usually wait till the dog is around three years old.
NO
if your dog has gone through a heat without getting pregnant in between pregnencies she should be fine
If the female is trying to get pregnant with a male dog then it will go on the males back but if the female dog is going on the males back while pregnant then it may means that they wanted to have another baby.
By mating with a male dog.
No
Sometimes.
Female animals do not begin to produce milk until they are pregnant. My suggestion is to talk to a local vet for a formula for the puppies.
doggy style. How did you think they do it?
Yes.
No. It's only during a specific stage of the heat cycle when a dog can become pregnant. Once the dog is pregnant, she won't have a heat cycle. Usually the cycle resumes within six months of having a litter of puppies. Three weeks after weaning a litter (usually when the pups are 5 weeks old) the female dog can be spayed, so she will no longer be able to become pregnant.