No, Sozin is Zuko's great-grandfather. Ozai is Zuko's father.
Sozin's Comet
Azula hits Zuko with a lightning bolt in "Sozin's Comet Part 3: Into the Inferno", which is the 20th episode of the third season, Book 3: Fire.
I know right!!! It never actually tells you which is really sad but at the same time it's nice to have some mystery left. In the last episode, Sozin's Comet, at the very end you see Zuko asking his father but it doesn't actually tell you, I guess Zuko never finds out either. Zuko became Fire Lord so he had the power. He was very strong-willed and would find out before the final scene.
in bato of the water tribe and sozin's comet part 2: the old masters
they have 2 fights. one was before Avatar started and you Iroh tells us how Zuko got banished and why zuko got his scar. when Zuko finally returns he faces his father on the day of black sun when Ozai cant firebend.
Not all the firebenders are evil. Some are though: - Prince Zuko - Princess Azula - Firelord Ozai - Firelord Sozin
Zuko became good and became Aang's firebending teacher. Then, when Aang defeated his father, he became 'Fire Lord Zuko'.
she wanted to be the favorite of their father so she could be the fire lord and not zuko.
We can assume that Zuko has already passed away, since The Legend of Korra is set seventy years after Aang's death, there is a chance that he didn't live long enough to see the reincarnation of Korra. Then again, this is only based off what season one has revealed. Katara is still alive in season one so on the flip side he could still be alive. We can only make guesses at this point because the show still hasn't revealed anything about Zuko.
You can only begin playing as Zuko in Into the Inferno when you get to the Boiling Rock level, when Sokka and Zuko set out to find Sokka's father. After that, Zuko will be one of the two different characters you can play as in each level.
I believe it's Book 2: Earth episode 9 that General Iroh teaches Zuko the lightning move.
On his father's side, Zuko had one cousin, Iroh's son Lu Ten. We don't know if he had any cousins on his mother's side.