Is the income derived from owning a live animal profit or rent? Natural resources are considered non-produced and yield rents, including land and the fruits of the land, e.g. trees. Does this extend to live animals? Or are live animals considered “produced” goods (capital) that yield profits?

Does it make a difference whether the tree or animal is wild or has been planted on a plantation or bred in captivity respectively?
Live animals held and raised in captivity (and any products created) are counted as production. In particular, they're counted as &quot;livestock value-added&quot;. From World Bank (2014):

Outputs from the livestock sector include the increase in the number of animals and the production of livestock products.
The increase in number of animals is represented by both fixed capital formation — i.e. animals that are inputs into the production process, such as breeding animals and adult males for breeding or animal traction — and by so-called ‘work-in progress’ animals, namely those reared for slaughter and young animals reared to become fixed assets.
Livestock products include meat, milk, eggs, hides and skins and other by-products, such as manure, hides and skins, fat, offal, honey, transport services, etc.

You also ask:

Does it make a difference whether the tree or animal is wild or has been planted on a plantation or bred in captivity respectively?

Yes. From UN System of National Accounts (2008, PDF, emphasis added):

Certain natural processes may or may not be counted as production depending upon the circumstances in which they occur. A necessary condition for an activity to be treated as productive is that it must be carried out under the instigation, control and responsibility of some institutional unit that exercises ownership rights over whatever is produced. For example, the natural growth of stocks of fish in the high seas not subject to international quotas is not counted as production: the process is not managed by any institutional unit and the fish do not belong to any institutional unit. On the other hand, the growth of fish in fish farms is treated as a process of production in much the same way that rearing livestock is a process of production. Similarly, the natural growth of wild, uncultivated forests or wild fruits or berries is not counted as production, whereas the cultivation of crop-bearing trees, or trees grown for timber or other uses, is counted in the same way as the growing of annual crops. However, the deliberate felling of trees in wild forests, and the gathering of wild fruit or berries, and also firewood, counts as production. Similarly, rainfall and the flow of water down natural watercourses are not processes of production, whereas storing water in reservoirs or dams and the piping, or carrying, of water from one location to another all constitute production.