Welcome to the forum.
What OS are you running?
The short answer is no.
The question is, why would you want to? The second monitor is typically used as either part of the desktop, to which you can drag and arrange your workspace as you see fit, or it takes the video feed direct but shows nothing of the desktop and workspace.
I used to do it with my BMD HD Extreme card, the SDI out would show the editing of the sequence timeline. Its great for finding faults like overscan issues etc. and for showing clients the edit as it is being put together. It also makes a much less cluttered workspace.
The other big thing, is that the SDI outputs all 16audio channels which I use a lot, as there is no way to create a file (that i know of) using all 16 chanels once completed. The most you can do is 5.1 (6ch).
The downside is that it only works when editing in a BMD sequence. Just today i have had to try editing in a different sequence as im having other quality issues which dont seem to be happening in the new sequence settings but are quite significant in the BMD settings.
I've dragged the program monitor screen onto a 2nd monitor like you have suggested, but i agree its far from ideal. I'd kill for my old setup where it is full screen output....
NO, But in the project settings you can select a playback option and chose the second monitor, and you will see the preview on second monitor full screen. If that what is you after.