I have Palatino Linotype installed on my Windows XP system and your sample epub displays fine with the current stable 32 bit Windows versions of ADE, Calibre and Sigil. What's your system configuration and what Sigil/Calibre versions did you test your epub with? Also check your Qt version by selecting About from Sigil's Help menu.
The only other thing that I noticed that you didn't enclose the font name in quotation marks.
IIRC, font names consisting of multiple words should be enclosed by single or double quotation marks and, as theducks
has already pointed out, you didn't specify a fallback font. Your stylesheet contains the following line:
font-family: Palatino Linotype;
which should read something like:
font-family: "Palatino Linotype", serif;