Unfortunately I dropped my phone (galaxy S3) and the screen is black, so I cannot do anything on the phone. But other than that, it is fully functioning, and I can see the files when connected to a pc. There is this question which states that the phone needs to be rooted in order to access the database where the contacts are stored. Is there an other way to retrieve contacts when the phone is not rooted? The gmail sync was unfortunately not enabled. Is it possible to turn this on remotely?
Can you use Kies to backup the contacts from the phone and access them on the PC?
Or maybe even the new SmartSwitch software that has replaced Kies, although I have never used it.
You can buy a 11-pin MHL to HDMI adapter and hook your S3 up to your TV. Since your touchscreen doesn't work either, you'll have to buy an official "Samsung" MHL adapter along with an OTG adapter so you can hopefully use a USB mouse along with the TV output. (Not tested)
Another idea with a guarantee of working, you can just buy the $2.00 OTG adapter to hook up a USB mouse, and use the hardware keys to take screenshots within 3 seconds of moving the mouse, then copy the screenshots onto your computer to check your mouse position before clicking. It will take a while, but you will have a couple bucks into it with 100% success rate.
Finally, you could just get the screen fixed.
Hope this helps