Marrying a first cousin by blood is the closest relative you can marry so legally it doesn't constitute incest.
Breeding with a close relative is something that goes on all the time - generally it is not a good thing because inherent genetical defects (even ones that are currently dormant) tend to be magnified or concentrated, while breeding with the tribe over the hill tends to dilute genetic defects...
I don't think medical science is developed enough to be able to predict more than a handfull of possible complications and it would no doubt be very expensive to have them make even the few possible predictions...
Such is the state of society for the last half century that few people who meet under any circumstances can be certain it is not a half brother, sister, father or mother or aunt or uncle they are bedding down with. A dna test would tell them, of course, but as it is an expensive and time-consuming process and not necessary by Law most people decide to do without...
The only incestuous relationships ever heard about are the very small percentage that go wrong and make the news or are brought to the attention of the authorities! Most of the royal families of europe have been almost incestuous for centuries and most of the offspring haven't been any different from the average...
Even ordinary people have historically found comfort in close relationships - Laurie Lee wrote in his autobiography of a brother and sister living together in his home village as man and wife "and they had two beautiful children"...
Relate, the UK relationship counselling organisation estimate there is more than 50,000 quite successful incestuous relationships being conducted in the UK at any one time. I don't expect they have all produced offspring but some of them must have done and if the results were seriously malformed I am sure it would have made the newspapers - and it didn't!
The advice that seems appropriate to anyone considering breeding with a relative is to look back at the family history as many generations as possible. If they all had a normal lifespan and didn't suffer from anything unpleasant and had relatively normal intelligence then I would consider most objections to be over-ruled! Any offspring will probably be fine and the chances of any problems is likely to be similar to the rest of society...