"am I just trying way too hard to somehow get my dog back?"
It sounds to me like your processing and going through the stages of grief.

A flaw I see in your rationale is " If we didn't exist, then we did, then we didn't, we're in the same place we were before we existed last time."

This is counting nothing as if it were something. If you did not exist there was no you not to exist was there?

Non existence by definition cannot be a criteria of existence. They are opposites.

How it appears to me is thus.
We have bio-hardware (brain and nerves), we have plug n play extensions (limbs, organs, senses), we have a power source (Stomach, Heart, lungs), exhaust (waste removal)
We have an operating system (autonomic nervous system)
AND
We have a software program runinng which is an evolutionary algorythm we call "I", "Me", "I think therefore I am"

I do not see how the software runs without its hardware.
I hope you feel better.

Circular logic is things go in circles because they do not go square.

Atoms are rather rounded even though they can make crystals in square or pointed shapes.

Stars, planets and moons are rather circular as well and move in a circular type direction.

Square atoms, square stars? I don't think so.

The point is, a starting point, that has possible spherical(circular volume) direction in every direction. To go back to a starting point would be that all possible directions are open to reoccur in even but it could go in a different direction (360° × 360° × 360° if the event were to happen again. Then, all of the other chain of event factors after the start would effect and eventually lead to some things different.

Splitting an atom with nuclear bomb then expecting to put the atom back together again with the same parts and energy it started with.

It seems to me, cognition, our thinking thought capabilities, is energy based and as we die out brains denature and the energy process that holds out cognitive ability disapate or disperse into the atmosphere around.

I have questions about could this cognition capabilities have ability to go with energy transfer. Kind of like an example of a radio station broadcasting.. A person speaks into mic, mic translates energy of sound into electricity that gets amplified thru the radio tower in radio waves then recieved by a radio and plays on a speaker.

Can our cognition stay together after brain dead to "ride" on other forms of energy?

I'm so sorry for the loss of your dog. When I think about reincarnation I often think about these people who say they were a queen or a king in a past life and I laugh because who says if reincarnation is true that we will be human again? We could come back as insects or any other species.

Your hypothesis makes perfect sense. Intuitively, I personally can't fault it.

X + n - n = x

The only thing missing is what is the independent variable that elicits change in x!

Nice work, Sergeant!

Scientifically speaking, "Energy can neither be create nor destroyed it merely changes form" as a Science teacher of note once informed our class.
So, logically, it should apply to matter as well since matter is merely a combination of atoms held together by energies of attraction and repulsion so when matter (material things such as our bodies, etc,) can longer sustain itself into the form it has taken as influences by these energies and chemical reactions, etc, then it too returns to or changes into a different form but is not truly destroyed simply because the remaining atoms and energies will be re-used, recycled into something different over the passage of time.
As to the 'energies,' for want of a better term, of memories, knowledge, etc, that were a part of the operation/s of our brain, then I tend to think that some of them become encoded somehow upon our DNA/Genes and are passed down to our descendants/offspring, hence what is loosely referred to as being 'Race Memories' or a kind of Dejavu occurrence from time to time.
Though some may choose to scoff at this, I have experienced a kind of inexplicable 'dejavu' occurrence quite a number times in my 60+ years of life, for example, in 1998 my Daughter and I were invited to Egypt to return a small statuette of an Ancient Egyptian person that my Great Grandfather had purchased in a Bazaar in Cairo after the Battle of the Nile.
He believed it to be a fake and merely passed it down through the family until it ended up with me and I proceeded to learn to translate Egyptian Hieroglyphics and found it was not a fake but a Commemorative Statue to honour Imhotep commissioned by the Pharaoh Djoser.
On being taken on very private tour of the Great Pyramid of Khufu I remarked to Zahi Hawass that I had a strange feeling that I had been here before and pointed out to him that below our feet ( we were in what is called the Queen's Chamber at the time) there was another passage way leading down even further into the base of the pyramid.
Zahi, looking quite surprised, said to me, " Have you been to Egypt before, you must have been BUT how did you know about the passage way and were it is, we only discovered it about 3 weeks ago and nothing has been published about it as yet?"
Both my daughter and I replied, almost in unison, " We've never been to Egypt before you invited us to come so how could we know something like that?"
It still puzzles me to this day.

What is your dog made of? Carbon, nitrogen, oxygen...your memories? Its behavior? Its DNA ?

By existence one is talking about a particular consciousness.Individual consciousness not comes from being a member of a certain tribe (species, culture, life experiences and on) but what an entity gets out of a set of experiences. Again, there are an infinite number of lives possible in the universe and to expect one life (consciousness) is to rob another of theirs.

3 1/2 years ago my mate died (she hated the word passed and would always ask: what did they pass - gas?). We both realized death is death and all life will yield to this fact. We agreed, outside of memories, we would go on and even seek another. Besides, why have another like the one one had before. Variety really is the spice of life.

Reincarnation is a load of bullcrap, you could have cloned your dog. [viagenpets.com]

Sucks to lose your dog, but don't let wishful thinking cloud your faculties. If we (or it) 'came back' in a different physical form with no memories of knowledge of our past existence then there's no meaningful way to say it would still be us. If I died and was resurrected with no knowledge of my previous life then I would just be someone else - there'd be nothing there that makes me me. And other lives are all around us, so what's the diff? Treasure your memories, get another dog, and make new ones.

Not to be critical, but your argument is nonsensical. Define your understanding of "Nonexistence: or your exact meaning as "coming from nothing". And it's the Law of Conservation if Mass, not energy.

By the same token that you talk about the possibilities of reincarnation, if there were truly nothing, then even this would not be possible. You derive from the merging of two gamete cells. One from your mother and on from your father. This "is", something, not "nothing". This unfortunately renders your premise specious at best, ridiculous at worst.

Nope.

Only one ride on the merry-go-round of life per lifeform.

Start out as a spark. End as a short circuit.

Get another dog. They all will love you.
.

there are varying degrees of consciousness. each little neuron pulse is a bit of information. the more neurons to fire and brain matter to capture some of it as memories the greater level of consciousness. 13.5 million years ago australopithicus shared a cranium size and content the same as all the other primates.

Shortly after the Pan group (orangutans) split off ... our shared primate ancestor suffered an abnormal copying on our first chromosome in a section that is unstable to this day. The area in question codes an enzyme that makes stem cells continue to multiply instead of specializing to split. It's known as the NOTCH enzymes I believe. Anyway this particular enzyme tells stem cells that reside in the cerebellum to not worry about specializing just yet. Go ahead and keep multiplying and you can specialize later (personifying the process that the little string is engaging). This makes the base number of glial cells that produce neurons greater, which means when they start splitting after specializing their density will be exponentially bigger. This is known as the NOTCH2NL enzyme or somesuch.

What had happened was a piece of DNA split and multiplied during replication ... and it mistakenly also created an abnormal duplicate. This happens quite often actually, especially in certain areas of our DNA which have relatively unstable structures - such as this particular spot on chromosome 1 that to this day is a genetic marker for autism and a couple of other neural disorders. So there was our regular NOTCH2 enzyme spot occupied by a perfectly normal enzyme maker PLUS an extra one that was inactive due to being faulty. At one point we shared this with all primates except the pans. This is fact. But about 4MYA or so human ancestor hominids "suffered" a curious genetic mutation. During a replication of the NOTCH2 irregularity the DNA corrected the area by an overlay. And then it also added TWO MORE such producers of the NOTCH2 enzyme. Now they are labeled NOTCH2ABC which gives us 4 such enzyme producing sites over the previous one. This made the base number of neuron-producing glial stem cells increase such that our neural density went up 6-8x. These are facts.(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6092419/)

Subsequent hominids had a huge increase in brain power which made consciousness increase exponentially as well. The mind was capable now of understanding the world with much greater detail and complexity. This is when man's evolution kick started. I believe all consciousness is just that awareness on whatever level of sensory input a creature is built to accept. Our inner "self" is the accumulation from moment to moment of all those senses interacting over time.

I personally believe that awareness is beyond time so the fact we have this illusion of a self thanks to moment by moment continuity ... means that once time isn't involved it exists in its' entirety or it doesn't exist at all. It's like a flip book animation. The animation only exists as a derivative of the moment by moment action ... so when time collapses IS there animation or is there not? Was anything created?? I say there was and it is the exerciser of free will. And that is consciousness. There's a lot of physics involved that I am tired of discussing so you can just believe it or not. My point is there does exist an avenue for hope scientifically - if you are so inclined.