Is the mutation rate in organisms in general consistent over the genome?
Well, for start, there are "mutational hot spots", regions that are more prone to mutation than others.

As for immune system genes, first of all, lung cells and heart cells and retina cells don't need to mutate those genes, because they don't use them.

But you are right that in immune cells there is a lot of DNA futzing in the sequences for the heavy and light chains, in order to generate diversity in T and B-cell receptors.  Those receptor sequences are built in a mix-and match fashion, like building an outfit from a closet with a few pairs of pants and a few shirts.  Add in some messiness in cutting out the specific sequences, and you have a fair bit of diversity in receptor sequences possible from a single genome.

Later, once a B-cell or T-cell meets a matching antigen, the cell multiplies, and each descendant cell is subject to somatic hypermutation, in the hopes that one of the tweaked sequences will better match the antigen.

But none of that is going to be very evolutionary relevant, since only mutations in gametes or gamete stem cells are passed on to the next generation.