Goten and Trunks were both born from very "strong" Saiya-jins, much stronger than the maximum a normal Saiya-Jin can be without transforming. Therefore, in order to handle the power they were naturally given, they could automatically change into the higher powered form of Super Saiya-jin.
Turning Super Saiya-jin changes your body genetically (it obviously has a profound effect on physical appearance). So, the gene to change Super Saiya-jin will always be passed on to any Super Saiya-jin's children.
It having extra characters who are half Saiya-jins would be useless to the already long Dragonball Z story, so Akira Toriyama just made it so they could naturally turn Super Saiya-jin without the pain of training and causing the extreme anger needed to transform.
When Future Trunks first appears, Vegeta doubts that he is a Saiya-jin because he has purple hair. He says that Saiya-jins have black hair.
Unexplained: Vegeta's hair was red-ish in the beginning of the Vegeta Series. I say that's what happens when you colorize a black and white manga. They just changed it mid-way, to look better.
In an episode, Vegeta says that a pure breed Saiya-jin is born with a hair style and remains with that hair style for the rest of his life and it cannot be changed.
Unexplained: In the Bardock Special, Nappa has hair. In the Vegeta Series, he doesn't.
Gokou and Vegeta are full breed Saiya-jin's and are born with tails. Gohan is a half breed Saiya-jin, but he also is born with a tail.
Unexplained: Why don't Goten and Trunks have tails? Some say that the tails were cut off at birth and others say that the offspring inherited the human parent's trait of not having a tail. I say the importance of the tail was lost when Gokou turned Super Saiya-jin. Since both Goten and Trunks can naturally turn Super Saiya-jin, they didn't need the tails.
When Team Ginyu is first introduced in the Freeza Series, Gurudo has a flashback on his encounter with Vegeta. The setting inside the flashback took place before Vegeta set out to invade Earth. Therefore, Vegeta is supposed to still have a tail. Vegeta's tail is missing in the flashback.
Nameks are classified into two groups: healers and warriors. Healers can heal anyone, but can't fight. Warriors can fight, but can't heal.
The Namek race was based on snails and slugs. This explains why Nameks can regenerate lost limbs.
Nameks live on nothing but water. They do not need to eat.Unexplained: When Kami told about his past growing up in the region where his ship crashed from Namek, a scene shows him eating fish.
Grandpa Gohan found Baby Gokou on Mount Paozu.
Unexplained: The Vegeta Series shows that Gokou was next to his space pod when Gohan found him. Bardock Special shows that Gokou was found sitting by himself in the forest when Gohan found him and that his space pod was behind him. The Vegeta Series also shows that Piccolo finds Gokou's space pod in the wilderness (away from where Gohan found Gokou) when he's training Gohan.
The guardian of the Earth split into 2 forms: Kami (pure good) and Daimoah (pure evil). Daimoh had a son, Piccolo, who fused permanently with Kami. Piccolo also permanently fused with Nail.
Unexplained: If Piccolo died, then Kami dies (and vice versa) because the two were originally one; but when Daimoah died, Kami and Piccolo didn't.
