How to maximize `Preview` by keyboard shortcut?
To make Preview open PDFs in the "Two Pages" mode, go to Preferences --> PDF, then select the desired mode in the "Show as" menu.

There is no shortcut to go to "Almost Fullscreen Mode" by default, but you can create one:


Go to System Preferences --> Keyboard --> Shortcuts;
Click "App Shortcuts";
Click the "+" button to add a new shortcut;
You'll be asked for the menu title you want to link to a shortcut. For the command you want, it is "Zoom";
You will also be able to specify whether you want the shortcut to be available in all application or a specific application only.


EDIT: Here's an AppleScript solution to set the first window of Preview (the frontmost opened document of Preview) to the size of the screen (works only with one display connected):

tell application "Preview"
    tell application "Finder" to set screen_resolution to bounds of window of desktop
    set bounds of window 1 to screen_resolution
end tell