Questions about the role of Azir

His first role is to be a mage, then the second, 'marksman'?

Why can Azir be an AD carry? (Is it about his skills or attributes)

To follow up, if I am to play him as an AD carry, where should I rely most on my attacks? On skills or on auto-attacks?

A marksman isn't necessarily an AD-Carry. It's a champion that can deal damage from a safe distance. Teemo for example is also labeled as a "Marksman" although his AD build isn't too viable.

So basically: Azir cannot (or at least should not) be played as a classic AD carry. He usually builds AP, CDR and sometimes Attack speed but his playstyle is similar to that of an AD Carry. You send out your soldiers and attack from afar.

"Primary role", "secondary role" and the bars displaying a champion's attack power, defense power, and difficulty etc, are mostly there to help new players get an idea about a champion.

I doubt they are meant to be accurate for ranked games. For example it would be a bad idea not picking an ADC like Jinx and instead getting Teemo, just because he has a "Marksman" tag on him.

In normal games, players tend to try out new things and not trying hard, so Azir Marksman might work. (By marksman i mean a build focused mostly on your autoattacks.)

ADC Azir would still be a bad idea since nothing scales with his AD.

Your soldiers cause magic damage when "you" autoattack. Your total damage would be mostly magic. Therefor, your primary type of damage would be magic. That means that you must build AP items, and perhaps get some item that increases your attack speed as well. A good fit for that would be Nashors Tooth, since it gives you AP, attack speed and cooldown reduction (which further increases your attack speed because of his passive).