Any genetic diseases that would prevent a person from developing an average intelligence?

Absolutely. Severe intellectual deficiencies have a very high frequency of genetic and/or biological causes, a growing number of which can be identified quite specifically. The most common known genetic source of mental retardation is fragile x syndrome.
Certainly, but rare. There are in excess of 200 metabolic pathways in humans where inherited errors can lead to life altering changes.A well known one is PKU, which if untreated leads to mental deficiency,seizures & many other symptoms.Many of these are rare, recessive disorders that affect 1/10,000 or less.Each of us are thought to carry some errors, which is why its bad to marry your siblings= bad genetic outcome.