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 $Y \to X$, you assumed $X$ was affine, but you don't need it for this). This condition is basically that the non-singular $X$ has rational singularities. In characteristic zero, this is a fairly straightforward application of Grauert-Riemenschnedier vanishing and holds as long as both $Y$ and $X$ are smooth (it doesn't matter whether you did blowups at smooth centers or not). Interestingly, we still don't know whether this vanishing holds for regular $Y$ and $X$ in mixed characteristic (although we do know this vanishing for regular $Y$ and $X$ in positive characteristic http://front.math.ucdavis.edu/0911.3599.
For the second question "fibres are not unions of projective spaces", you've already seen one non-normal example. Ulrich gives a nice example of a non-rational component in the comment below (originally I said something stupid here).
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.