Animals most definitely go to heaven, because they are living souls just as we are.
And yes, it is ok and proper to pray for a beloved companion animal, and even to anoint them with water or oil. Animals are also God's loving creations, and He loves them very much.
The best word to say are the ones that are on your heart. Simply speak to God and each other.
You and your neighbor and her dog are in our prayers.
In the Love of the Lord,
Frank and Mary