I'm so sorry to hear about your sweet dog's passing. Having also been through it a few times with old, dearly loved pets, I feel your pain (((Hugs))).
We recently lost a relatively young cat (nearly 4 years old), struck by a car. Thankfully, he did not suffer and died from a glancing blow to the side of the head, so we were able to spend some time petting him and come to terms with the reality that our sweet kitty was no longer with us. We wrote notes and drew pictures to put in his box, and it was a beautiful, warm, sunny day when we laid him to rest. One of the children made a delicate wreath of fine vines and tiny purple flowers and a daffodil which we laid on his sweet head. We talked of special times with him as he lay in the warm sun one last time. He rests under a heart-shaped rock topped with daffodils next to two other beloved kitties (who died of old age).
It's the first time I've lost a pet so suddenly, and it's been a hard loss. He was a unique and special cat, well loved by our seven children, and we're all grieving his loss daily. I served as 'midwife' at his birth and the birth of his 5 litter-mates (we still have them and the mama cat, but he was the character with an extra dose of sweet in the bunch), and I was the one who gently laid him in the place where he rests now.
Do I believe we will see him again in eternity?
Yes I do.
I don't believe that God, Who places such value on life, just ends innocent life. Animals (yes, bugs, too) were affected by the Fall, but bear no fault for it. The more I learn of God's character, the personalities that He's put not only in pets, but in wild animals too (who hasn't seen on YouTube deer and dogs playing, the relationships that elephants and other animals have with each other - one of the fish in our aquarium and one of our cats play 'cat and mouse' on a regular basis, etc.), and the undeniable bond that He's allowed to exist between people and animals who come under their care - it just doesn't make sense that He would just erase them from His creation and from our minds and hearts.
God is all about redemption, restoration, and LIFE.
The Ecclesiastes passage? It comes under the category of 'wisdom and poetic literature' in the Bible. Like those in the Psalms and Proverbs, the writer asks questions, ponders, and makes observations about mankind, the world we live in, and God Almighty. Does Psalm 10 tell us that God stands far away and hides in times of trouble? No - the psalmist is asking the question.
Other declarative Scriptures reveal to us the character of God - Truths in which we can rest when the questions come.
I don't know the details of how eternity will be, but I do trust God and His love for me, His creation, and His love for life. And based on Who He is, I trust that our loved ones who are in Christ, as well as our pets who have passed on, are safe in His Hand and that we will be with them again.
Grace, peace, and comfort to you in Christ,
-JGIG