There are some people who automatically assume that low IQ is related to having mental disorders. While this can be somewhat true because those who have learning disabilities may not show the same progress as those who do have “regular” learning abilities, having low IQ does not automatically mean that a person is mentally ill.
The results of the IQ test is not always accurate. There are some who have answered it for fun and without thinking too much. Some people were careless and some just did not want to follow instructions. The personality of the person taking the test may also be a factor in this.
No. Mental disease, by which I imagine you mean disorder, can occur to anyone irrespective of their standing on an IQ test. However, if by 'low' you mean so low that any score on such test would be meaningless, i.e an adult person who functions like a three-year-old, then, yes, it is possible that a birth defect has caused poor physical ability and health issues as well as low intellectual functioning.
If you mean 'low' on the bell curve of normal distribution for intelligence, that is with mild-moderate learning difficulties, then there is no reason why their mental health should be poorer than those scoring higher. There are certain syndromes, such as Down Syndrome, where physical abnormalities are present together with low intellectual capacity, but such individuals would not be described as suffering from "mental disease" and typically they are happy and positive people who bring joy to their families.