You share (about) 1/128 of your genes with a third cousin. The chance of any (particular) recessive trait being passed on from both parents as a result of the inbreeding is a whopping 1 in 65,536.
I'm not sure what you computed. The interesting question is to pick a recessive trait from one of the parents and ask for the chance of it being passed on. The probability that it's being passed on from the parent who has it is 1/2. The probability that it's being passed on from the other parent is (1/2)*(1/128). The combined probability is 1/512, not 1/65,536.
I'm not an expert on this, so perhaps you could tell me if a family history of cousin marriage increases the likelihood of the recessive trait being passed by both parents. I mean, if for the last 10 generations a family has been marrying cousins to each other and introducing little variation from people outside the family, the odds are increased, right? I'd imagine cultures with a history of cousin marriage, such as the Pakistani community in the UK, are much more prone to this type of problem than when cousins marry, but none of their forbears married a cousin.
Giving birth after 35 (in this case 37) is statistically more risky than offspring from first cousin's. A third cousin is much more distantly related than a first cousin, your first cousin has a pair of the same grandparents as you, your third cousin shares a pair of your Great-great-grandparents.
