dog mating

Question
does the amount of times that dogs mate during a females heat determine
the sex of the puppies or the size of their litter?

Answer
No.

The number of puppies will depend on how many eggs are ovulated by the female, and the ones that actually get fertilized by the sperm will determine the size of the litter.  

Sex is determined just like in humans with the sperm carrying the deciding factor X=girl Y=male.