How do mutations actually occur?
Mutation of DNA can be caused by an exogenous or endogenous source. Many external factors like irradiation or chemicals induce mutations. Some endogenous mutations arise from oxidative stress, insufficient DNA repair or spontaneous mutations on the molecular level.

There are many mechanisms for DNA repair like base excision repair, nucleotide excision repair, homologous end-joining or nonhomologous end-joining (Löffler, Biochemistry and Pathobiochemistry).
But it is more difficult for the cell to repair the mutation if it is present in both DNA strands. 
According to your second question:


  DNA polymerase could just skip a section of the DNA or add in extra bases


DNA polymerase cannot reconstruct what was the original strand or the orientation of the strand. To prevent cell death, some DNA repair mechanisms include alteration in the genome. The DNA polymerase can reads through and the mutation will be replicated when the cell divides. Consequently, the key question is how the effects of the mutation influence the ability of the cell to survive.