Mongol diet was heavy on dairy and meat, since they're primary food sources were from grazing animals which they could easily bring with them in their nomadic wanderings. Every Mongol boy was trained to hunt as well and so a great proportion of calories would come from game caught on the fly during Mongol journeys.

They often ate horse products, especially the prized mare's milk, sometimes prepared by being cultured like a yogurt or fermented into an alcoholic beverage. They also ate other ruminants' milk and meat.

The cavalry would typically have satchels of dried or smoked meat which could be easily eaten in the saddle, which enabled long drives on horseback without resting. Actually the constraint on their drives was allowing their horses enough time to graze. They rarely prepared meat by cooking over fire when on military advances, which also enhanced the ability of Mongol raiding parties to remain undetected until too late.

They prepared meat by drying it when they could in preparation for long campaigns or the riders would put a satchel of raw meat from recently hunted game under their saddle on long trips and the constant impacts would tenderize the meat enough to make it more digestible. They certainly cooked meat when time and circumstances allowed but I don't know the relative prevalence of these types of preparation.

Since the Mongol diet was so high in fat and low in carbohydrate/sugar, they were able to go long periods without eating or eating very little due to the high energy density in fatty foods. There is some speculation that this led to them being in better health than the agricultural peoples they conquered, but it is uncertain whether that is really due to the diet rather than the extremely physical lifestyle.

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***EDIT because I realize I didn't really answer this part:

And did the Mongols suffer from scurvy or anything related to have very few vegetables in their diet?

Mongols ate all parts of the animals they killed. They would eat the innards first because those couldn't be dried or stored well. If all internal organs were consumed one can get all necessary vitamins and minerals, including vitamin C which is apparently present in great quantities in the adrenal glands (I just had to look that up) but also present in other glands and organs. However cooking destroys vitamin C, so the Mongols were eating these organs raw to get their full compliment of vitamins.

Another gross factoid is that the main tea drink of the Mongols was always generously contaminated with cow or yak manure, since that was used to clean the cups and pots, and sometimes added straight to their traditional tea, called block tea. Ruminant dung is simply fermented grass, so I would imagine they were getting inadvertent doses of vegetable nutrients that way.

Also, there are accounts of mongol children raised in the traditional way who interacted with western researchers and were found to have basically non-existant dental hygiene, yet reasonably good teeth, which the researchers attributed to the lack of sugar and starch in the diet. So there's another potential health perk from their diet.

I haven't read Jack Weatherford's book, so I don't know how good a case he makes, but the basic issue I have with this: I doubt the Mongols had a very different diet from other nomadic steppe peoples of the age, such as the Tatars, the Cuman-Kypchaks, the Jurchens, etc. Yet the Mongols typically conquered these enemies as decisively as they conquered settled agrarian societies. In fact, I can't think of any steppe nomads who were able to resist the Mongols as well as, say, the Vietnamese.

For that matter, the Han dynasty's people were probably eating something fairly similar to what the Song dynasty's subjects were eating, while the Xiongnu probably ate something not too different from what the Mongols ate. Yet the Han beat the Xiongnu, while the Mongols beat the Song.

Likewise, I wouldn't attribute the vast conquests of Alexander or the Romans to the Macedonian or Latin diets.