For myself, property starts to get really problematic when it is amassed by a particular person or group of people - this could include land and housing (land owners/lords), the means of survival or production (bosses, capitalists), or ideas (patent holders, etc.) If a collection of people live in a house, that is their house. Once they are gone, it is not. Theoretically at least (and in my experience of communal living situations), once a person leaves that house, it is not theirs anymore, it is still that of the commune (I have been trying soooo hard to avoid referring to "the commune," but in this case it is the best terminology I can think of).
Another piece of property as critiqued by anarchists is the way in which it is accumulated through heredity. So I own a house, and your dad rents a house. As it now stands, when I die, I can give my house to my child. When your dad dies, he can not give "his" to you, because, although he lived there, someone else owns it. That means that my child starts out with a house, whereas you are still beholden to a landowner. Over time (and especially in hierarchical and capitalist societies) this allows more and more property to be concentrated in the hands of a few. So instead of paying rent, my child can buy some more land or a business, where perhaps they employ you so you can pay rent, but at the end of the day, my child benefits from this more than you, even if he is benevolent, even if he pays himself equally to you, his employee, because when he dies, my grandchild now has a business and home, where as your child has no home (or maybe you bought one, but you are still a step behind), and no business.
As far as the trespassing questions, I dunno? depends what they are doing? I get a little ornery when randos come traipsing across the land I live on (though actually mostly that is because it sends my dogs into a fit of barking and rambunctiousness), but I am not going to attck them for merely passing through. On the other hand, if I looked out at my garden and saw some fool helping themselves to what I'd grown, I'd probably be really pissed off, though I also recognize that this is contradictory to some of my beliefs. But is this my ideal world? No. That is where this question is of interest to me.
Obviously we don't live in a world of free communal relations; most of the land on which (and homes in which) anarchists live are owned by others. In the context of existing now, it might make sense for some to "own" property, and it probably makes sense to defend that property, but this is certainly not our end goal, which would likely be, at least in part, to expand the commons (as Taigarun has defined it). It also in many cases probably makes sense not to own property, as it can (again, in a capitalist reality) become like an albatross thanks to things like property taxes, bills, and so forth.