Technical virology questions about ERVs and RNA vs. DNA viral rates of mutation?

This is stemming from a sidebar from two other threads as to not hijack them. I have two questions. The first is about endogenous retroviruses (ERVs). I understand they are compelling evidence of human evolution as their viral DNA is inserted in multiple places across hominoid ancestors. How does a virus insert itself into genomic DNA?

The second question is somewhat related. RNA viruses mutate/adapt faster than DNA viruses. The extent of my knowledge currently is that the DNA virus undergoes a “proof-reading” mechanism and the RNA is able to circumvent. What is this “proof-reading” mechanism, and what is the general mode of translation for each of these types of viruses?