There aren't many grown here, because too often winter will damage what would have bloomed the following year. Everblooming roses will usually have blooms on them all season, with more in early summer and fall. In addition to how often the rose blooms, you may want to take into consideration how long the blooms last and the number of blooms at one time. The quality and quantitiy of blooms on some recurrent roses make them as good a choice as any everblooming rose. I heard shrub roses have less problems with insects and disease.
All roses vary as to how susceptible they are to insect and disease problems.
As a group, hardy shrub roses tend to be more resistant than others, but they are not immune. No matter what type of rose you are growing, an ounce of prevention is worth a pound of cure. Given the right circumstances, even the most resistant plant can succumb to disease. The new Flower Carpet series being introduced over the next few years has earned several awards based on disease and insect resistance. It may have a few aphids or a little black spot, but it seems to be able to avoid heavy infestations or damage.