Depends. Depends on how functional and how severe the case is and how the perception of reality is with mom. How medically stable and how much support systems the child has.
Suspiciousness . the child may avoid others. The child may be hallucinating or just not make sense when talking and the parent might get frustrated. The child may be very disorganized and lack social skills. Any time you think something is wrong with a child go to a doctor and other psychologist to find out what is going on and what they suggest based on your circumstances, money and insurance options.
Can be totally fine. All depends on whether the child received adequate, consistent parenting either by mom or others. Having a mom who has mental illness is not the relevant question, and not the main issue that bear on likelihood of success in future relationships. There are many more people who end up having relationship difficulties and yet have perfectly healthy parents. Quality of parenting is more important.