There is not much to say, because you answer yourself. :-)
A family of Enriques surface is as you define it. The equality in 2) follows from Grauert's theorem.
The only thing is that Grauert is not enough to prove that $R^0f_\ast O_Y = O_T$, since it would only imply that $R^0f_\ast O_Y$ is a line bundle. But in characteristic 0 the equality $R^0f_\ast O_Y = O_T$ (for a projective morphism of Noetherian shcemes) is equivalent to the fact that the fibers are connected. One implication is [Har III.11.3]. I cannot find a reference for the other implication now, but it follows in your case from Stein factorization [Har III.11.5].
EDIT: I misquoted the result. One implication is always true, but the requires stronger assumptions, in particular $f$ should be flat. See Emerton's answer for a correction.