Do human cells contain only one copy of the genome?
As a quick answer, yes, most human cells carry 2 copies of the genome and are known as diploid cells. One copy comes from each of your parents, so they aren't identical, but usually pretty close. Sperm and egg cells only carry one copy of the genome and are known as haploid. During fertilization the 2 cells merge their copies and make a diploid zygote. At the chromosomal level, humans have 23 chromosomes, so a diploid cell has 2 copies of each so a total of 46.

As far as repairing damaged DNA, using one chromosome to repair its damaged counterpart is known as Homologous Recombination.