Will a child with encopresis get sick more often because of the constant contact with feces?

Almost Never. This problem is not related to to the immune system. Unless the feces is available to open skin wounds, there is generally no risk. 1 exception might be a urinary tract infection with fecal exposure to the penis or vagina and poor cleaning technique.
Not necessarily. Although leaking poo is a gross issue, I have not seen an issue of related illness in three decades of following these patients. Their biggest issues seem to come with the sadness of being different until they can gain empowerment thru treatment. Recurrence over many years is common, and sporatic (viral or other) diarrheal illnesses frustrate everyone involved but long term sucess is possible.