In addition to the answers that clarify that Zuko indeed is Ozai's son, I'd also add that to me it always seemed that the conflict Iroh describes as a fight between Zuko's bloodlines is basically him putting Zuko's inner moral conflict into a metaphor that makes it graspable. Both his powerful ancestors stood for a different moral view and that same fight is ...
Ozai is the father of Zuko, his mother only stated the opposite in the letter because she suspected Ozai was reading them and wanted to confirm if that was the case or not.
So the advice is still relevant.
No plot hole here, Zuko is very much Ozai's son.
It's revealed further in the comic that Zuko's mother, when sending the letter to her former lover saying they had a son, was deliberately lying. The goal was to force Ozai into admitting he was reading her mail.
Her former boyfriend would know it was fake as they never had sex.
The Avatar Wiki page on lavabending has as it's first Trivia point:
Avatar Extras for "The Avatar State" stated that lavabending was a combination of earthbending and firebending and thus something that only the Avatar could accomplish. Book Three: Change rectified this and revealed lavabending to be a rare specialized technique of earthbending.
As Mor Zamir's answer states, no one can bend more than one element, and the only reason the Avatar can do it is because of the spirit Raava.
However, as this question speculates, it might be that certain specialised forms of bending may be possible due to having heritage of different nations; specifically that lavabending might only be possible for a child ...
No. In order for an individual to hold more than one element, he/she needs Raava to hold it for him/her, as we learn in season 2, episode 8: Beginnings, Part 2
Also, we saw a couple of examples where parents from different nations had children who could only bend one element:
Mako and Bolin's father was from the earth nation and their mother from the fire ...