I've mostly read on the military aspect of nomadic peoples, not the societal, so perhaps someone else is better qualified to answer and can be more detailed than I am.

The short version, though, is that horse meat was definitely consumed by nomadic peoples, but it'd surprise me if it was the majority of subsistence. You have to remember that nomads are, in the first place, herders. They are pastoral nomads because they keep livestock.

Also, a big reason for their success in warfare was their expertise in archery from horseback. They learned this while fighting each other, to be sure, but also very important in this is hunting. Nomads, as far as I can tell, hunted a lot. Hunting was also an important way for post-nomads (ie the Mughals, the Qing) to stay connected to their nomadic heritage in the hopes of keeping their martial prowess intact.

So, yes, horse meat was eaten, but I would be surprised if it was their main food source, especially when at peace.

This is one of those posts I find hard to source, because it's sort of spread throughout the books I've read on nomadic warfare. The comment on Post-Nomadic dynasties hunting to maintain their nomadic roots is detailed in Gommans, Mughal Warfare. I'm not 100% sure, but I imagine I got some of this knowledge from Rossabi, The Mongols, A Very Short Introduction. The latter I can definitely recommend to anyone interested in the Mongols, as it's a cheap and handy book of about 100 pages with a LOT of information in it, written by an expert of the field.

As /u/Czone noted, horse meat likely served as part of the Mongol's diet while on the move, though it is unlikely that it was the main staple of it. As herders, they'd bring with them a variety of animals.

In On The Tatars, Ibn Al-Athir writes

Moreover they need no commissariat, nor the conveyance of supplies, for they have with them sheep, cows, horses, and the like quadrupeds, the flesh of which they eat, naught else.

Unfortunately I'm away from my bookshelf and can't cite anything else from memory, but fermented mare's milk is still produced and drunk in Mongolia today, and we can assume they would have also made dairy products from the rest of the animals they brought with them while on campaign.

Their diet while not on campaign was presumably as varied as the lands they conquered, but again, I don't have access to anymore sources at the moment.

On the Tatars http://sourcebooks.fordham.edu/source/1220al-Athir-mongols.asp

Side question: If nomads ate animal and animal products, am I right in assuming they do not farm? If so how healthy is their diet? Do they consume fruits or vegetables to supplement meat and milk? Wouldn't a lack of variety in their diet be detrimental to their health?