I believe that reincarnation, as a repeated physical materialization, happens at all levels of evolvement of consciousness.
A soul is an entity that chooses to dedicate a fragment of itself to materialize in the physical reality, to experience, learn and grow. It can simultaneously send multiple fragments, not only successively (here "successively" isn't related to the "physical time").
As I understand this, the difference between man and animals is that man mostly mastered instincts, is dominated by emotions, develops an intellect, has an incipient intuition, while animals are dominated by instincts, develop emotions, have an incipient intellect. Men are distributed on a bell curve centered between emotions and intellect.
Anyway, as I see these, we don't reincarnate, but whatever incarnated as us keeps reincarnating, until it graduates. This is differently from saying "we reincarnate".
Anything that lives and houses a spirit reincarnates as souls are not limited to form or only take some forms exclusively plus there is documented evidence of people having had animal incarnations etc even back and forth in some cases. Reality really heavily favors recycling souls by any and all means with some requirements though rarely discussed.
The way I understand the garden of eden story a large number of souls (not all, but this group would include us) did something that caused us to have an awareness of good and evil. Adam is a highly evolved soul saw that we needed guidance so he volunteered to guide us to a fuller understanding of good and evil. God looked at all the lifeforms on earth (tree of life) and there was one particular species that would provide us with the best biology to learn(made coats of skin). H sent angels to keep us from eating from the tree of life so our souls will never incarnate as animals.
There are probably other souls who are different from us or didn't eat from the tree and it's possible they could incarnate as an animal. I don't think biology needs a soul to enter for life so with regards to whether a pet will continue after death depends on weather a soul chose to enter the body.
It is my understanding that the process of reincarnation is in fact the evolution of consciousness (the soul) itself. This process takes millions if not billions of years for soul to become self-aware and develop an ego. So, everything in the cosmos is in fact apart of spirit.
Soul starts off in the mineral kingdom as a one-dimensional state of awareness. From there is will go to the plant kingdom as a two-dimensional state of awareness. After millions of years, it will evolve into the wild animal kingdom as a three-dimensional state of awareness. The first three stages are what is called a group state of consciousness. Meaning that once the body physically dies that soul returns to the group state from which is belongs to taking with it, its experiences while on the physical plane. This process is painfully slow from our perspective. As the consciousness evolves so does the physical forms change along with the inner subtle bodies of the astral and mental planes. However, at some point the soul will develop the ability to become self-aware to a degree while in the wild animal kingdom. At this point the soul will make a large leap in evolutionary consciousness as the lower ego develops and reaches the human kingdom. Here the soul becomes self-aware.
From here on out the both the physical, mental and astral bodies over many lifetimes become more refine and grow as soul starts to make its way back upwards to the higher spiritual dimensions from which it came. Once soul reaches a state of spiritual enlightenment it with take with it all of the experiences it as ever had over the course of its soul journey within the lower worlds of duality, time and space and cycle off the process of reincarnation at least on the physical plane.
It will only come back to the physical plane as a spiritual master, teacher, avatar ect.. to aid and help raise the vibration and consciousness to those still here on this earth plane school. There for everything in the cosmos is inner connected to spirit.