If the organism is too efficient at preventing mutations - it would evolve slower, right?

Yes, but note the mutation rate of germline cells - that are passed to your offspring and hence influence evolution - is estimated to be two orders of magnitude lower than other (somatic) cells.

Not necessarily if it had recombination (as in sexual reproduction), but as far as I understand, yes, you’d probably get fewer novel alleles/coding sequences of DNA generated per organism replication