This is a good question but you have also asked several questions (each of which varies wildly from empire to empire). I will, however briefly, address the one which I think you are most interested in.

My area of expetise is the 4th to 5th century European Huns (sometimes also called Attila’s Huns). So I will address this from that background. Thompson, 1999, and again Kim, 2013, point out that the Huns were more interested in the inheritance of people than of land. This is, I think, one of the core difference between nomadic and agricultural empires before the former conquers the latter. Before the Huns arrived in Europe, they moved like a snowball across the inner-Asian steppe, gethering peoples and growing in size but were not permanently tied to one specific place. It is easier the see the differences between this mobile, nomadic empire and, say, the permanently settled Roman agricultural empire.

When the Huns arrived in Europe and began conquring the local agricultural kingdoms/tribes, and then eventually even Roman Pannonia, they too then adopted and occupied their towns and cities. Eventually the Huns built their own permanent camps, the likes of which can be seen in Priscus' fragmentary history (Much as the Mongols turned Karajorum into a city). Additionally, for the first time, the Huns took up permanent residence around the Carpathian Mountains. Thus, the building of cities by nomadic peoples is not only brought about by influence from agricultural peoples but also by becoming tied down to a more permanent geographical region.

So, if nomadic empires can have cities and if they can begin to substitute large portions, if not their entire diet on agriculturally obtained foods, then what difference is there then between agricultural and nomadic empires once the latter settles? This answer simply can't be generalized across all nomadic states as they are too different. But for the Huns they maintained their highly mobile and nomadic lifestyles and turned that into a military asset. The same can be said for the Xsiungnu who, as Barfield argues, China had such a problem dealing with because of their mobility and could often only rival at great expense to the state. Ergo, I would argue, the symbiotic relationship between man and horse emphasized in the nomadic cultures of Mongolia, the Inner Asian steppe, and eastern Europe (during the time of the European Huns), along with the field armies of said nomadic states being able to transferform that symbiotic relationship into a military asset, is one of the more defining features of an urban nomadic empire.

Hope this helps.