Mutation rate in viruses
Really interesting question.
I am not sure, if the mutation rate is something "actively done" by the virus. I think its more a byproduct of vast replication after taking over the host cell. Mutations with a negative effect are deleted, those with neutral effects will be there, and those with positive mutations will be selected for. This paper looks interesting in this context: "Viral Mutation Rates". Looking for related publications on the site, I found this two:


Rates of evolutionary change in viruses: patterns and determinants.
A short introduction to the origin and molecular evolution of
viruses.


The whole list is here.

Answering your second question: There is obviously no connection between the mutation rate of the virus and the host. I have found this paper ("Rates of Spontaneous Mutation") which gives some pretty interesting numbers. In short: The rate of mutation for DNA viruses is much higher than for eukaryotic cells and for RNA viruses 2-4 orders of magnitude higher than for DNA viruses.

At least RNA viruses encode for some own polymerases, the reverse transcriptase. Depending on which these have a proof-reading (exonuclease) function they induce more or less errors.