How can prayer help me as a caregiver?

Absolutely. A recent conchrane reveiw on prayer andsurgery failed to conclude that prayer can improve medical outcomes, however, there is evidence that it can affect you as a care giver and the relationship you have with patients. Read the article in the american journal of surgery march 2011 on this topic (there is a link on my profile).
Pray always. From praying (communicating with god or a supreme spiritual being) you can get motivation for your sacrifice. You can remember how others have coped with similar situations while you pray , reading spiritual books (your preferred sacred book bible, coran or alike) or while you meditate. There is some scientific evidence that meditation changes the physiology of the body against stress.
Depends how you mean. Two sides to this. 1) Causality in prayer by its very nature is unprovable and therefore not documentable. Faith = belief w/o proof. 2) Prayer fulfils an emotional need for some people. The psychological benefits of prayer have been scientifically documented. Whether the object of prayer is real or not is immaterial. It's the act of prayer that effects the psychological benefits.