To what extent did the Mongols use non-Mongols in their army? Particularly non-nomads, because I know they hired/incorporated various steppe peoples.
And did these auxiliaries fight 'like Mongols'?
The Mongolian history is too long to be described in one post. Even now there are the Mongols and there is a Mongolian army. But usually when people speak about
'the Mongols' they mean the time of their unbelievable breathtaking conquests.
So in my post the time period will be AD XIII century. Somewhere in between 1206-1294.
To what extent did the Mongols use non-Mongols in their army?
There is an old copybook maxim that the British conquered India with the hands of the Indians. Though we know there were the British fighting and killed in India that saying is true.
Something similar could be said about the Mongols - they conquered the world with the hands of the peoples of the world, especially the nomads.
So one thing for sure - extent of using non-Mongols was enormous (though you won't find any exact numbers or percentages).
they hired/incorporated various steppe peoples.
Incorporated - definitely yes. To a great extent.
Hired - definitely no. At least in the 13-th century.
The Khan ordered - warriors went to war. Simple as that. The Khan was not supposed to pay his nomad warriors.
various steppe peoples.
And did these auxiliaries fight 'like Mongols'?
For an outsider from 'the civilized world' it was very difficult to distinguish between nomad Mongols and nomad 'non-Mongols'.
But ye, they fought 'like Mongols'. There was nothing too revolutionary in Mongol steppe warfare.
1. as technical specialist in charge of siege machinery and other tecnical equipment.
The Mongols had a lot of siege machinery. Of course most of the personnel responsible for that was from sedentary conquered peoples. The Chinese were widely used for that purpose. But in China the Mongols used more advanced type of mangonels from the Islamic world. The common sense suggests that the guys from the Arabic world made and operated these mangonels.
2. as 'cannon fodder'
Some non-nomads were used on a permanent basis. Thousands of warriors from Rus were taken to conquering the Southern China. As for as we know they never came back home. Something tells me that the guys from Rus were not the only sedentary folks who died far away from their homes fighting for the Great Khan.

Some non-nomads were used for a single campaign or a war. At least these guys had a descent chance to come back home. So the Russian contingets took part in the Mongolian conquest of the Caucasis. The Armenian troops were usually taken by the Il-Khans to fight against the Mamluks.
* It is funny how an Armenian historian proudly states that the Mongols so highly thought about the valour of the Armenian warriors that they had the honour to attack the Mamluks as the first wave. Of course the Mongols considered these Armenian guys dispensable (they did not care about their losses in manpower and fatal casualties). And in battle it was always a good idea to have unreliable troops in front of the 'Mongols' to control the situation.