Folders work the same way: right-click, choose Show View Options (or cmd-J). You get the same options, too: change grid spacing, icon size, name position, etc.
Make the icons small enough and names are only one line. Enlarge the icons and you get two lines.
Lines are longer with names positioned on right, for a given grid spacing and icon size. For example, "2011 06 22 June TRES & TRGY Board Minutes, Draft 3 - to be distributed" will show completely when positioned on right, but not positioned at bottom, for text size 13 + icon size 128x128 + grid spacing max. Smaller icons or smaller grid spacing and it won't all show.
In Finder windows, hover the mouse pointer over a name and it'll show a yellow box (tooltip) with the whole name. Also works in all view styles of app Open dialogs. Must hover over the name, not the icon.
The View Options for a folder don't affect the Icon view in an app's Open dialog. There may be a way to set those view options, but I don't know what it would be.
If you have the Open dialog open in an app, you can go to Finder and drag either the window or one of its files onto the Open dialog, and the dialog will go there. Since you have more control over Finder window views, it may be easier to customize the view for a Finder window and use it for navigation, and when you want to open a file, either drag the file onto the app's Dock icon, or drag the file onto the app's Open dialog.