Does Psalm 145:16 teach that G-d ensures the well being of every living animal?
"You open your hand and satisfy the desire of every living thing"
Yes, the major commentator Radak says this includes animals. Not well-being, though, as you ask, but just food.
This edition of Miqra'ot Gedolot lists the following commentaries:
Rabbenu Avraham ben HaRambam writes in HaMaspik L'ovdey Hashem (Wincelberg translation) that these verses apply to animals as well:
Just as He created each being of the world, so does He supply its nourishment, as it says "He gives bread to all flesh" (Tehillim 136:25)...It also says, "The eyes of all look to You and you provide their food in its time" (ibid 145:15). This applies not only to people, but also to animals. (page 271).
However, it is not clear (at least in the translation) whether he is discussing each individual species, or each individual animal. Indeed, one could perhaps infer that he is discussing the species, for he writes that "Just as He created each being of the world, so does He supply its nourishment". God created each species, but did not (directly) create each individual organism.
Rambam famously writes in Moreh Nevukhim (3:17) that personal providence does not extend to animals.
Accordingly, it seems reasonable to conclude that Rambam (and presumably his son) would understand this as referring to whole species; not to individual animals.
See also Rabbenu Avraham's discussion of this verse on page 277.