What if Piccolo and Kami never split into two separate beings? 

Kami splitted into two because his heart was evil and desperate to became Guardian of earth. He tried to put an end to this evil but he can't due to that Kami spiltted into two Kami the good one and Piccolo the evil one. If Kami killed Piccolo, Kami would die with him because Piccolo was the half part of his life if half part dies then other half part will die too.

Elder namekian said that Goku would be super saiyan if he defeated Kami. But What if Kami didn't spiltted?

Kami wasn't kind hearted, Being a Guardian of earth needs to be kind hearted or the death of old Guardian of earth so if the old Guardian of  earth dies then Kami will become the Guardian of earth. Lets assume that old Guardian of dies somehow and Kami becames The Guardian of earth. 

In Dragon ball Goku wanted to became stronger because he wanted to defeat piccolo but piccolo isn't there, but Goku wanted to became so he goes to Kami with the help of power pole, He defeats just like in series. But Kami is stronger than the super saiyan due which Goku cannot defeat Kami. Kami wasn't much evil hearted that he won't let Goku surpass him so he would make him stronger and stronger. That thing still won't make him super saiyan but, maybe like in Freiza saga the fake super saiyan, However It still won't defeat him.

See also: Namekians vs Saiyans

Goku still wouldn't became super saiyan But will be tough for Raditz in Dragon ball z. Goku would die fighting with Raditz (because of the tough fight both dies). Vegeta would be shocked that a lower class warrior defeated Raditz because of it vegeta and nappa would come to earth and it was their mission to destroy earth. Kami would take him to king kai but he still ins't match for Kami so after training with king Kai he would eventually go to Kami to defeat him but cannot defeat him and loses again and again Due to sake of Earth goku would become desperate to defeat Kami and due to it Goku would became super saiyan and Defeat Kami.

Goku would Kill Vegeta because Goku is to powerful for Vegeta and also due to Vegeta's pride. The series will go without Vegeta and also Vegito and Gogeta.

Related: What if Goku Killed Vegeta?

In Android Saga Future Trunks helped Goku with the heart virus But in this Goku killed Vegeta so Future trunks won't come and Goku dies due to heart virus.