Q1: It is the other way round. For a smooth family the differential is surjective and the relative tangent is the kernel, so you have an exact sequence
.
In this way the tangent to actually restricts to the tangent of the fibers.
Q2: I don't think that the classes are determined by the fibers alone; they depend on the family. It does not even make sense to say that are determined by the fibers since these classes live in anyway, so you have to know at least the total space.
But since restricts to the tangent of the fibers, you know, by naturality of the Chern classes, that if is the inclusion of a fiber .
And these you can compute using the fact that is Enriques. Namely since twice the canonical is trivial and .
Q3: Surely it is not injective in the top degree, for trivial dimensional reasons. I do not see any reason why it should be in other degrees.
Q4: As is written in the article, this follows from . This is more or less clear in cohomology. In this case is the integration along fibers, and since is times the fundamental class of for all fibers (see Q2), that integral is .
To translate this in the Chow language, I think the folllowing will do. Let be a cycle representing . Since is smooth, we can compute the intersection number . So intersectts the generic fiber in points, and the morphism has degree . In follows that , which is what you want.
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