Is Lord Slug guru brother?

It is revealed that Lord Slug is the other half of Super Kami Guru. Ironically, Guru is the evil half. While Slug is the other half of Guru, the latter also considers him a brother, as when Frieza calls Guru a "slug", Guru responds with "Leave my brother out of this!" referencing Lord Slug.

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How did Goku go FSSJ?

Having no more options, Goku's anger surged, causing him to gain a massive power increase, manifesting his Super Saiyan powers and making him become like a Super Saiyan, enough to quickly overwhelm Slug. Are half Saiyans stronger than Saiyans? Yes, half saiyans are significantly stronger then their pure counterparts. This is simple because they contain a much larger potential compared to the pure blood saiyans. "So why isnt gohan, trunks, or goten stronger?"

Consequently, who is the god of all saiyans?

Yamoshi (???, Yamoshi) is an ancient Saiyan who could transform into a Super Saiyan long before Goku could. After his death, his spirit wandered in search of six pure-hearted Saiyans, to create the ceremony that would birth the first Super Saiyan God. Consequently, what is the evil saiyans name? "Evil Saiyan's name is confirmed to be ????/Kanb?, which is likely meant to be from the latter half cucumber. There's no real reason to be using "Kanba" over Cumber."

Can half Saiyans use God Ki?

As Akira Toriyama explained it. A half Saiyan can channel Yamoshi's God ki with the aid of 5 other pure hearted beings with some Saiyan blood, or can absorb it from a Super Saiyan God's ki augmented strikes as Vegeta did, if he has a pure heart.