Beliefs among Christians vary widely.
Most Christians agree that All of creation is beloved of God.
Most Christians do not believe that animals can sin. Sin requires (by most accounts) intentional action of a type of which animals are not capable. Furthermore, many accounts of sin will reflect the idea that sin is an injury to the person's relationship with God, but while animals are thought to be loved by God, they are not thought to be free-willed creatures and thus have a different type of relationship than human beings.
Most Christians who believe that humans go to heaven or hell when they die do not believe that animals are also sustained in eternal life in this way. However, some look to scriptures that talk about disorder in the natural world as a result of sin (Romans 8:22) and suggest that in the Incarnation Christ came to redeem "all flesh" which includes all living things. Also, some Christians believe that eternal life will not be elsewhere, but will be life in this universe/world/earth once God's plan has come to fruition and the effects of sin fully overturned. (N.T. Wright is easy to read on this version of the Christian afterlife). It is easy to see how this idea of life after-life in this world would continue to include animals,... but in a new way as suggested by the biblical image of the lion laying down with the lamb and eating straw. (Isaiah 65:25)
As to the claim that human beings are responsible for all other animals, this is a matter of much debate. What does such stewardship include? Is it dominion (responsible for, but not possessors of) or domination (use animals any way you want)? There is too much to say about this matter.
Some example literature... On the one side: Andrew Linsey is a Christian who believes in robust animal rights.
Charles Carmosy is a Roman Catholic Christian who describes our responsibilities in anti-speciesist terms like Peter Singer's crowd.
On the other side: Matthew Scully argues that God grants humankind almost unlimited power over animals, but desires their compassion for creatures.
https://www.amazon.com/Creatures-Same-God-Explorations-Theology/dp/1590561422/ref=sr_1_1?s=books&ie=UTF8&qid=1489510552&sr=1-1&keywords=Andrew+Linzey+animals https://www.amazon.com/Love-Animals-Christian-Ethics-Consistent/dp/1616366621/ref=sr_1_sc_2?s=books&ie=UTF8&qid=1489510512&sr=1-2-spell&keywords=Charles+carmosy https://www.amazon.com/Dominion-Power-Suffering-Animals-Mercy/dp/0312319738/ref=cm_cr_arp_d_product_top?ie=UTF8