(Of course, you have the implicit assumption that the equation of degree $d$ is not $0$.) Yes, for The answer is yes. In the following reason: 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/ . I'll look up a more Your special case is essentially Cor. 2 on p. 152 of Matsumura's "official" reference soonCommutative Algebra". To get to the general case, if I have a chanceuse 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.
(Of course, you have the implicit assumption that the equation of degree $d$ is not $0$.) Yes, for the following reason: the "slicing criterion for flatness on the source". $\mathbb{P}^n_Y \rightarrow Y$ is flat. See Exercise 25.6.F in the May 12 2012 version of http://math216.wordpress.com/2011-12-course/ . I'll look up a more "official" reference soon, if I have a chance.