I agree on both counts. IIRC pure Mongolian armies had 10% "heavy" (melee) cavalry and the rest horse archers. The dividing line could be a bit fuzzy, by the way. The Mongolian nobles that made up the heaviest cavalry would also have used bows; and while I am not sure if the Mongols employed light lancers, these would have counted as melee cavalry.
Going into central Asia brings you into conflict with the Seleucids, and if you attack them it´s likely that everybody else stops doing so, so you get to face the Grey Death alone. Going east also brings you into contact with other nomads and their lands are just not as profitable.
As for the third question: I don't know much about the Sarmatians, but nomads in general do not have a concept of an organized military. Every man is a warrior, so having a standing force does not make sense. Obviously, you can have a concentration of warriors at a disputed border, but these would be local tribesmen defending their own land.
For that matter, nomads don't have much in the way of government. They don't need to, because population density is low and every tribe can and has to look after its own survival. The steppe is a harsh environment and nomads pretty much spend the entire year preparing for the winter. This breeds hard people that are mostly concerned with their own survival. They do not have any sympathy for those who cannot defend what they own. For the same reason they are very fierce about protecting their own territory and possessions: they cannot spare anything, and any sign of weakness may induce neighbouring tribes to take your land and your goods. These neighbours too live on a knife's edge.
The only reason for the tribes to accept a King of the Tribes is the threat of force (either from the King's tribe or an outside power) or because banding together allows them to extort or plunder neighbouring cities. Even then, the tribes would mostly look after their own affairs, and it would be an unwise monarch who tried to take a tribe's supplies from them without compensation. Taxation is not appreciated by people who believe they can look after themselves.
That should also answer your fourth and fifth question. Out of necessity, nomads are opportunists. They don't go to war to prove their masculinity or superiority, since survival on the steppe is the only measure of worth. They do not want land, other than as pasture for their herds, since they cannot spare people to colonize it. They don't care about freedom: on the steppe, everyone is a slave to the environment.
This also means the tribes cannot afford to lose many men; nor are the individual warriors going to risk their lives more than they have to. Nomads go for targets of opportunity. They want the wealth (or supplies) to survive the winter, and they'll happily take protection money over plunder if attacking would be risky.
Role-playing is going to get tricky once you control many settled areas, as this usually is the point where the nomads realize taxation is more profitable than looting. This means taking over the administration of the previous rulers (since nomads don't possess such a thing) and settling down as the new aristocracy. It will also result in the confederacy falling apart, since the original conqueror is usually dead and there is no concept of hereditary monarchy. Every ambitious chieftain fancies himself a king.
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