Do I believe animals will be in Heaven? Absolutely. According to my biblical hermeneutics, Heaven is going to be like it was in Eden.
And there were certainly animals in Eden.
Also Isaiah 11 v 6, about the messianic age (return of Christ) says " And the wolf will dwell with the lamb, And the leopard will lie down with the young goat, And the calf and the young lion and the fatling together; And a little boy will lead them. Also the cow and the bear will graze, Their young will lie down together, And the lion will eat straw like the ox.…"
It certainly depicts animals in Heaven. I'm not sure if it is literal or not, but I believe it to be.
I also think it's significant to look at the story of Noah and the flood, which in many ways was a picture of the coming destruction of Earth and then creation of new Heavens and Earth. In the first destruction (through flood), Noah was required to spare some from each kind of animal. It is fascinating to speculate on what this might mean for the days yet to come, but if nothing else it teaches us something about the heart of God.
God looked at the animals after He made them and said they were good. Then He went to great lengths to spare them instead of just smiting them off the earth and starting from scratch. The continuity was important to Him. And I believe it will be still at the end of this age.
Do animals have souls?
That is more of a complex philosophical question, and also a theological one. I'm not sure that it is terribly consequential though. The way I see it, if animals do have souls, then surely it is in the character of God that he would destine them to eternal bliss rather than eternal punishment because they are not sinners. And if they don't have souls, then He can reproduce our exact pets in Heaven for us and they would be no different from the ones we knew on Earth.
So basically, yes. I believe we will be able to see our pets in Heaven. I believe God delights in our love for them and wants us to celebrate that love for eternity, and He will make a way for that to happen.