If we look at the transcript we can see that Ozai has stood down from the throne and decreed that his daughter will become Fire Lord. Given that he's abdicated, from a legal perspective he's no longer the Fire Lord which neatly answers the first part of your question;
Cut to the plaza, the location of the Fire Lord's coronation. The
scene cuts to the front of Azula, who is kneeling down on the plaza
steps. She is seen wearing Fire Lord robes with her armor underneath.
Standing behind her are Fire Sages, one begins to hold the headpiece
over her head.
Fire Sage : By decree of Phoenix King Ozai, I now crown you Fire Lord [Azula]
The wider question (whether Zuko has the right to the throne) is slightly trickier.
It's pretty clear that a member of the royal family can ascend to the throne by killing or incapacitating the current Fire Lord. Iroh talks about his reluctance to challenge his brother and simply being seen as the latest in a long line of usurpers;
Iroh : Even if I did defeat Ozai, and I don't know that I could, it would be the wrong way to end the war. History would see it as just
more senseless violence, a brother killing a brother to grab power.
The only way for this war to end peacefully is for the Avatar to
defeat the Fire Lord.
Later on, Zuko challenges Azula to Agni Kai, a fight to the death. Given the lack of complaint from those present, it seems that from a legal perspective the winner will have the right to the throne;
The Fire Sage is just about to crown Azula the new Fire Lord,
regardless of what Katara and Zuko have just said, until Azula signals
with her hand for him to stop.
Azula : Wait. You want to be Fire Lord? Fine. Let's settle this. Just you and me, brother. The showdown that was always meant to be.