No. Great question with some interesting aspects. For instance, if a first child is autistic, parents may choose to not have others, which can look like first children are more at risk. Lots of research on birth-order effects, but most of it is not about mental illness so much as personality traits, job choices, that kind of thing.
Only Child. Unless you have genetic vulnerability of environmental predisposition, most children should have a normal life, children are usually very resilient.
No strong data. There is some debate about such things in the psychoanalytic literature, but I'm not aware of any strong evidence that sibling configurations lead to any predictable patterns of mental health or illness.