Yes, He does!  The Bible is filled with references of God's care and provision for animals.  These references include direct commands as well as symbolic references such as Jesus' parables.  It is clear that God gives separate attention to the animals. We are doing God's work when we care about animals.

The Ten Commandments instructs not only that men and women should rest on the seventh day, but also cattle (Exodus 20:  8-10)

The Heavenly Father feeds the birds of the air (Matthew 6:26-30)

God quenches the thirst of the wild donkey (Psalm 104)

He causes grass to grow for the cattle (Psalm 104)

The high hills were created for the wild goats and the cliffs for the rock badgers (Psalm104)

God had compassion for the animals in Nineveh (Jonah 4:11)

During the creation, God gave every green plant for food for the animals (Genesis 1:30)

After the flood, God made a covenant with man and the animals (Genesis 9:9-10)

God instructs that even if we don't like our enemy, we must care about their animal (Exodus 23:4-5)