As with many questions about Egypt and Mesopotamia, it heavily depends on when. Because you mentioned Ur and Lagash, I imagine you want to know about the Early Dynastic Period (2900-2000, but mostly 2500-2000), since these cities will no longer be the center of Mesopotamian politics after the invasion of Sargon of Akkad.

The third millennium is a fascinating time. We start having our first royal inscriptions, cities grow much larger, we see edicts, law disputes and other texts that allow us to try to understand the Sumerian fabric of society (I say Sumerian as a synonym for "South Mesopotamia", things were different in the north - Babylon, Sippar, Eshriunna). It is the tale of the Palace and the Temple.

The Sumerians would use the word "é-gal", which pretty much means "big house", to describe two large buildings in their respective cities. We name them "Temple" and "Palace", but it would be a mistake to imprint our conceptions of palace and temple into them. The Temple existed before that time, and it was the religious center of any city. They had increased in size and role during the fourth millennium, becoming effectively the administrative centers of cities. They would provide much needed social technology for Mesopotamia: the ability to muster workforce to projects larger than an individual family, or tribe, could conceive. In fourth millennium Uruk the temple (the Eanna) was rebuilt over and over again on the same spot, always increasing in size. It would not be like a Greek temple, with a huge open space with a statue of the god, these temples had rooms, corridors, store units, they were big houses with a central space for worship (reserved to the elite priests). We can see the transition of the temple from a central people gathering point to a central resource gathering point and eventually a resource distribution and production center. This evolution was probably very natural, the temple ended up with functions that today we see in a church, a mayor's office, a tribunal and, perhaps most importantly for our discussion, a factory.

Temples were huge production powerhouses and responsible for the majority of the output of non-agrarian items in a Sumerian city: craftsmen would produce using resources gathered by the temple and these products would be sold by the temple. They are also indirectly the major producer of agrarian items. Having the social technology to build great projects, the temple will be responsible for building canals, irrigation and to reclaim fertile land from the vast marshes of lower Mesopotamia. This land would belong to the temple, and they will hire workers, temporary of permanent, to work their land. Even after the emergence of the "Palace" in the third millennium, a secular administrative building in the city that would house the "king" (quotes are to remember you that attributing modern conceptions to these terms is wrong!), the Temple would remain the most important production building and land owner in Sumeria. So to your question: "Did all landowners live in town", landowners were mostly the Temple, so yes, they would hire other people to work their fields. It is important to note that most land did not belong to a person. Privatization of land would take some time to happen, and even then the structure of the temple would still be extremely important.

But did farmers live in cities? We believe so, yes. During the four millennium we see a large growth in size of Sumerian urban centers. The rise of these centers led to the abandonment of the surrounding countryside. Uruk, for example, grew 70 hectares between 3500 and 3200. This concentration of population within its walls led to the disappearance of nearby villages in a 15 km radius. This means that all arable land around Uruk was farmed by people living in Uruk, temporary of permanently. They did not own most of the land and were payed in rations by the temple according to their age and gender. Given the structure of the Sumerian economy, most likely the majority of the urban population mid-third millennium worked in farms or in food processing by the orders of the temple.

The social structure defined by the temple in the fourth millennium had two different levels: workers from inside the temple and workers from outside. From the inside you have administrators, craftsmen, merchants and priests, while the outside workers were food producing. This distinction will become social stratification in the third millennium, and solidify a class structure that will define the Near East until... well... almost today.

By the end of the third millennium we see the model of the Temple being copied by other "large houses". The Palace will also increase its production capacities, and it would specialize sections of it (the King's Big House and the Queen's Big House for example). Other smaller "big-houses" will eventually appear, formed by groups of families, and this would be the beginning of private ownership in Sumeria.

On a final note, this write-up makes it easy to believe that these notions were static. Mesopotamian economy was very fluid between agriculture and pastoralism, between nomadic and urban life. There are times when cities dwindle while there are others when they boom. There are farm settlements set up by the temple in reclaimed territory, and there are small villages in low-yield regions that suffered from the rerouting of the canals up-north. The theme of the "city versus wild" is very much in the minds of the Sumerians, and it is one of the main themes of the Epic of Gilgamesh.

Source: The Ancient Near-East - History, Society and Economy by Dr. Mario Liverani. Do not buy the Routledge edition of this book, for the love of Marduk. The paper used in this edition is extremely thick and the font size is way too small. Put that in a large book, you have the impression of reading clay tablets yourself. It is very hard to comfortably read this book, as the paper thickness makes it heavy and the font size makes it unreadable in most positions. There is a severe lack of maps and references to places, especially when Dr. Liverani lists ten different cities and regions that no longer exist to make a point about a particular socio-economic interplay. There are not enough timelines, maps and the caption and legend to many figures is just insufficient to even comprehend the point the image is trying to make. Instead of a map of Ebla in the chapter "The rise of Ebla" we see a photo of an aerial view of Tell Mardikh, where ancient Ebla was located. This is useless. In the whole chapter you have no idea where Ebla is located, only for brief references like "eastern Taurus and northern Zagros". Instead go read Van De Mieroop, M. (2008). A history of the ancient Near East.