(1) A family of hypersurfaces of degree $d$ over an affine scheme $Spec\ R$ means the following: it is a closed subscheme of $\mathbb P^n_R$ given by a homogeneous polynomial $f(x_0,\dots,x_n)$ of degree $d$ satisfying the following condition:
Every fiber is a hypersurface of degree $d$. This means that for every prime ideal $m\subset R$, the reduction $\bar f\in k[x_0,\dots,x_n]$ is a polynomial of degree $d$, where $k=R/m$.
Now, let $m$ be a maximal ideal, so that $k$ is a field, and look at the graded ring $k[x_0,\dots,x_n]/(f)$. For each $a\ge d$, the degree-$a$ part is a vector space of dimension $\binom{a+n}{n}-\binom{a-d+n}{n}$. Thus, some $\binom{a-d+n}{n}$ monomials can be written as linear combinations, with coefficients in $k$, of the remaining monomials. This is done by solving a system of linear equations obtained by setting $x^m f=0$, where $x^m$ are monomials of degree $a-d$.
Now, consider the same system of linear equations with coefficients in $R$. For each of the $\binom{a-d+n}{n}$ monomials as above you get a principal minor $M$ of your matrix, and the reduction of $\det M$ in $k$ is not zero. Thus, over an open set $Spec\ R[1/\det M]$, this determinant is invertible, and the monomial can be eliminated.
Thus, over an open neighborhood $Spec\ A$ of the point $[m]\in Spec\ R$, the degree-$a$ part of the ring $A[x_0,\dots,x_n]/(f)$ is a free $A$-module. Recalling how $Proj$ is covered by $Spec$'s and that a free module is flat, this implies that $Proj\ S[x_0,\dots,x_n]/(f)$ is flat over $Spec\ S$. (We will assume $R$ and so $A$ to be Noetherian here for simplicity.)
This is the main trick for proving flatness over a non-reduced base: you prove freeness instead. For a finitely generated module over a Noetherian ring, flatness and freeness are equivalent. So for a projective morphism $f:X\to Y$ a coherent sheaf $F$ on $X$ is flat over $Y$ iff the sheaves $f_* F(a)$ on $X$ are locally free for $a\gg0$.
In (2), you are mistaken about Hartshorne: Theorem III.12.11 (Cohomology and Base Change) has no assumption for the base to be reduced. So if $H^i(X_y,F_y)=0$ for $i>0$ then $f_*F$ is locally free, for any (Noetherian) base $Y$ and coherent sheaf $F$ on $X$, flat over $Y$.
For higher direct images, $H^i(X_y,F_y)=0$ for $i\ge i_0$ implies that $R^{i_0}f_*F=0$. But you can have $H^i(X_y,F_y)=0$ for $i> i_0$ and $H^{i_0}(X_y,F_y)$ non-constant, and still have $R^{i_0}f_*F=0$ (compare the Poincare line bundle on $A\times A^t$, as in Fourier-Mukai).
Suppose that $Y$ is the spectrum of a smooth curve over perfect field $k$ and let
$D\subseteq Y$ be a finite set of closed points. Let $f:X\to Y$
be a proper morphism and let $D\subseteq X$ be a normal crossings divisor. Suppose
that $f$ is semi-stable relatively to $E,D$ and $k$, in the sense of
Illusie in par. 1.4 of "Réduction semi-stable et décomposition...", Duke Math. J. 60 (1990).
The morphism $f$ is then flat and lci and its fibres are reduced normal crossings divisors. There is a relative residue sequence
$$
0\to \Omega_{X/Y}\to\Omega_{X/Y}({\rm log})\to F\to 0\ \ \ \ (*)
$$
where $F$ is supported on the singular locus of the singular fibres of $f$, and $\Omega_{X/Y}({\rm log})$ is the locally free sheaf of differentials with (relative) logarithmic singularities along $D$.
See for instance p. 23 in "Une conjecture sur la torsion..." by V. Maillot and D. Rössler
(Publ. Res. Inst. Math. Sci. 46, no. 4 (2011) - for lack of a canonical reference (?)).
Now let $M$ be any quasi-coherent ${\cal O}_Y$-module.
The tor-sequence corresponding to $\otimes_Y M$ when applied to (*) gives
$$
\dots\to {\rm Tor}^1_Y(\Omega_{X/Y},M)\to{\rm Tor}^1_Y(\Omega_{X/Y}({\rm log}),M)\to{\rm Tor}^1_Y(F,M)
$$
$$
\to \Omega_{X/Y}\otimes_Y M\to\Omega_{X/Y}({\rm log})\otimes_Y M\to F\otimes_Y M\to 0
$$
and since ${\rm Tor}^l_Y(\Omega_{X/Y}({\rm log}),M)=0$ for all $l>0$ (because $\Omega_{X/Y}({\rm log})$ is locally free and $f$ is flat) and
${\rm Tor}^l_Y(N,K)=0$ for any $l>1$ and any quasi-coherent ${\cal O}_Y$-modules $N,K$
(that is because $Y$ is the spectrum of a Dedekind domain and any finitely generated quasi-coherent
${\cal O}_Y$-module has a two-step projective resolution; the general case follows from compatibility of Tor with direct limits), we see that
${\rm Tor}^l_Y(\Omega_{X/Y},M)=0$, for all $l>0$, ie $\Omega_{X/Y}$ is flat over $Y$.
EDIT As remarked by Liu below, the sheaf $\Omega_{X/Y}$ can also be seen to be flat simply because it is the subsheaf of a torsion free sheaf.
Best Answer
(Of course, you have the implicit assumption that the equation of degree $d$ is not $0$.) The answer is yes. In the case where $Y$ is locally Noetherian, it is true by the "slicing criterion for flatness on the source", as $\mathbb{P}^n_Y \rightarrow Y$ is flat. See Exercise 25.6.F in the May 12 2012 version of http://math216.wordpress.com/2011-12-course/ . Your special case is essentially Cor. 2 on p. 152 of Matsumura's "Commutative Algebra". To get to the general case, use the general technique that finitely presented morphisms (as yours is!) can (locally on the target) be pulled back from the Noetherian situation (see Exercise 10.3.G in the notes linked to above); but this may be more than you care to know.