Screenshot paste only works in some apps, not in Google Chrome
I've seen this kind of behaviour with other functionality, try the following command in Terminal (may require sudo)

Change directory to "/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/"

cd /System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/

./lsregister -kill -seed

OR

/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -kill -seed  

This rebuilds the launch services database, sometimes corruption in this can cause weird behaviour



EDIT:
Since the above didn't help let's try something else.  When you use shift+control+command+4 to copy a section of the screen to the clipboard you can view the clipboard contents.
To do so, switch to Finder and click Edit > Show Clipboard to show the contents of the clipboard.  You should see something like this window pop up that says "Clipboard" at the top and has a description of the clipboard contents in the bottom-left corner.  Do you get the same thing?