As pointed out above in the comments by Piotr Achinger, I'm going to assume you meant the map to be proper.
For the first question then: sure, the fact that $R^i f_* O_Y = 0$ for $i > 0$ implies your first assertion that $H^n(Y, O_Y) = H^n(X, O_X)$ (here $f$ denotes the map $Y \to X$). This condition is basically that the smooth $X$ has rational singularities.
For the second question "fibres are not unions of projective spaces", you've already seen one non-normal example. I believe that fibers of such maps should always have unirational irreducible components (unirational means birational to an image of $\mathbb{A}^n$ under some map) but nothing more. I'm sure one can cook up normal examples too, but I don't know of any with the possible exception of the following counter-example to your third question (which I haven't checked).
The last question, "is it always true that $H^n(Y,O_Y)=0$?", here's an example which this time is normal. It is Section III of Cutkosky's A new characterization of rational surface singularities Note the $R^2 h_* O_Z \neq 0$ in the papers notation is exactly the non-vanishing you were interested in.
Note: In the interest of full disclosure, this paper was pointed out to me by Hailong Dao in this answer on math overflow.