The concept of "ownership" is a human perception that is imposed upon a noun. Ex. This is an apple vs. This is my apple. The transition from the apple being simply "an apple" to being "my apple" indicates that a subjective change of the human mind has occurred, not an objective change of the apple.
we can understand "own", as a fact of property, even it is not the objective fact, that if I own a thing, that it is objectivly mine. The word "own" can have a functional pragmatical meaning. If someone own a thing, he has some rights for it. "own" indicate this function.
This is untrue, property is a real thing that can be observed in nature. We can show that one action is the property of another, e.g. a leaf only exists because a tree nourished it, body motion is the property of the nervous system, etc. A beehive exists solely because of the ingenuity of bees
You seem to be defining the term property as a "characteristic", rather than my intended definition of property, which is "something that is owned". Also, each of your examples are over-simplifications. Ex. There are many reasons why a leaf exists, perhaps most importantly because there is sunlight.
You can not separate the characteristics that are unique properties of objects from the idea of ownership. The "properties" of anything are labeled as such because it is observed to "belong" to a specific case. Your course of action, when you remove the apple or nourish its life, is your "own".
An apple or leaf is both owned and the property of the tree until it is combined with the properties of another. Similar to how an apple or leaf is not owned by sunlight, but it is the result of the properties of the sunlight.
Just because an object has unique characteristics does not imply that said object "owns" said characteristics. The characteristics of an object simply represent how said object exists in nature. I understand your point, but I believe there is a distinction between "property" and "characteristic".
Ex. I would not say that I own the characteristic of myself being short, but rather, that being short is a characteristic of how I exist. One cannot own the characteristics of shortness, hardness, slowness, etc. These characteristics simply just "are" or "exist" with respect to a given object, and are not "owned" by said object.
The terms that imply ownership (my, mine, yours, theirs, etc.) are subjective representations. The statement "this is my apple" is an opinion. In order for the above statement to be objectively true, it would have to read as "this is an apple that I use". "To own" = opinion, "to use" = fact.
They aren't subjective terms. If they were, then the thought of who owns something would change based on perspective. Property disputes are a way of expressing differing viewpoints, but this doesn't invalidate the concept of ownership, e.g: The squirrel feeds her baby from her nut cache
The fact that you are able to assign actions to ownership such as the extent to which are using something is proof that ownership can be objectified. You can try to deny this process exists, but the only outcome is making claim to no property, or claiming that everything must belong to no one.
A lion's catch
Rodent's hole or cache
Ownership is a subjective consensus but is still real. Love is also subjective but real. Example: 
: If you have Persons A & B, about to enter a black hole's event horizon and A has an apple, A&B are discussing whether to trade the apple for a sum of money. The Apple is kept out of the event horizon by rocket as the two fall in. From an outside observer ownership lies not in the apple but inside the minds of the two. Information cannot escape in a readable fashion from the event horizon - therefore knowledge of whether they completed the trade and hence who owns the apple does not exist in the universe accessible to the apple or observers of the apple. There is no Objective information anywhere on the apple indicating it's ownership.
that means you cannot own yourself or your organs - meaning I can come along with force and take your heart and organs out with surgical equipment and use them for something else, scientific study, to help a loved one of mine with a transplant - even though you'll die I have as much right as you do
Unfortunately, I would agree. One can not realistically own the organs within a body, but rather, one simply uses the organs within a body for means of survival. Human rights are agreed upon in order to protect ourselves from the crude possibility of organ harvesting for capital gain.
Your body was given to you as a gift at birth, and you maintain it as property. You do own the organs of your body, what you are describing is a violation of property. The fact that someone can use your body without your permission is proof that protection of property is a biological imperative.
If one can not truthfully own something, then one can not evaluate the extent to which said something is owned, because said something can not be owned to any extent. In other words, there is no "more" or "less" with regards to owning something, for said something can not be owned in the least.
false, because ownership is subjective and given by other people to that person - so if a group decides that their friend has ownership of X in their eyes, then in the set of those people it is true that the person owns X and ownership is possible. you cannot mandate against ownership unilaterally
in a ontological standpoint: there is not ontological natural (science) fact, that I owe my organs. The organs in this view are just `in me´. Ownership in this sense in not something subjective, it doesn`t even exist. Or you say, that we have a ontological natural right over our body...