In general I think it would he highly unusual, if not totally unheard of, for an actual individual monk to inherit real property (unless you were talking about a powerful layman holding the abbacy of a monastery, but even in the periods when "monk" remained a fairly fluid category that's not really a monk). You *may* also find examples of irregularities like this in the earliest periods of western monasticism (say, before Charlemagne), I'm not sure. However, that doesn't answer your broader question, "how separate from the world is the monk, really," which is a complicated and interesting matter.

The answer will vary considerably depending on *when* in the Middle Ages you're looking at, and also depending on order, individual house, and particular individuals/situation. Some orders placed much more emphasis on really finding a deserted, out-of-the-way area to build their houses in (Cistercians in particular being associated with this preference and often breaking virgin soil for their houses and supporting lands) while others (Cluniacs and Benedictines in general, which are both more ambiguous categories than "Cistercian") were often very closely intertwined with the territorial nobility. In this latter case, though monks did not inherit, it would be quite common for a particular family to be traditionally "allied" with a particular house over several generations, a relationship that included donations of land to the monastery, sending men from the family to be monks in the monastery (where they may occupy positions of authority and use their connections, lay and monastic, for the benefit of the family, the monastery, or both) and even leasing land from the monastery to lay members of the family.