What are Azir's Soldiers?

Solution 1:

They are somewhat special. Think of it as an Untargetable Zyra plant with all the negative effects of auto attacks.

The soldier itself is definitely some form of untargetable and invulnerable minion (As I said like an invulnerable Zyra plant or heimer turret).

Now the attacks of the soldier are a mixture of auto attacks and spells. The player himself has to command the attacks directly by clicking on a target in range. Note that your Azir will stop moving if he is out of range but the soldier is able to attack the target while he will just walk up to the enemy target to attack if there is no soldier near. The fact that this is triggered by a right click (or attack move) against a target implies that they count as an auto attack in terms of minion aggro.

But then again the attack originates from the sand soldier and not from Azir himself. This allows it to be a spell that applies on-spell effects (Hello Zyra Plants).

TL;DR
The Soldiers are untargetable minions, The attacks count as Auto Attack for Azir and as spells for the soldiers.

Solution 2:

http://leagueoflegends.wikia.com/wiki/Azir#Sand_Soldier

Sand Soldier is a controllable minion that will autoattack in Azir's place when in range of the target enemy.

• Basic attacks are lines area of effect meaning they will affect all enemies in the same line as the primary target across their full range. Spear passthrough range: 50
  • Basic attacks will apply spell effects, counted as AoE.
  • Basic attacks will not be blocked by Spell Shields.
  • Basic attacks do not apply on-hit effects.
  • Basic attacks cannot be dodged, parried or blocked.
  • As Sand Soldiers are untargetable, they cannot be blinded.
  • Because there is a brief delay between the command and the actual attack, basic attacks are dodgeable.

This text is quoted from the wiki

By dodgeable (last sentence), they mean that the player can avoid the hits by moving and by cannot be dodge they mean from abilities like Jax E

Solution 3:

I would say that they are considered spells. The only thing I don't understand is why his soldiers can destroy wards? They can't destroy Teemo shrooms (as you said) but can destroy wards with soldiers just like an auto attack. As far as I know he is the only champion to be able to destroy wards with a spell (although champions like Renekton can destroy a ward in one go when using Ruthless Predator. I don't know how much of this was intentional or whether it's just something going on in the code that wasn't originally planned.

Having had a quick look through the wiki I discovered this:

"His soldiers'attacks don't count as auto-attacks and will not receive any benefit from auto-attacks. It still however procs the passive of the Furor boot enchantment."

It seems as if it's some sort of amalgamation between auto attack and spell. Having both the benefits/downsides of both at times.