1st Answer: Most likely in the sequel, but we (fans) don't know for sure.
2nd Answer: Probably, because it was rumored that Neytiri (the girl) was supposed to be pregnant in the end of Avatar, the first movie, but James Cameron decided to cut that part out. So possibly. I hope so! :)
3rd Answer: If humans were designing the avatars genetically from the ground up, they would most likely make them sterile to avoid any complications.
4th Answer: Jake's avatar and Neytiri are significantly genetically divergent. Even if it were possible for them to have a child and carry it to term, it might not be healthy.
5th Answer: If Jake's avatar and Neytiri have one or more healthy children that themselves can have children, human DNA could spread throughout the tribe and alter their genetic makeup significantly over time. This is a serious ethical delimma.