1. If there is a ground wire available in the box, it will probably be bare uninsulated copper, or have green insulation, solid green or green with a yellow stripe.  If the wiring is run in metallic conduit to a metal box, the box itself may give you the ground you need.  

2. You can't connect to the ground for another branch circuit, but for a retrofit like this you may be able to run a ground wire to the ground rod or other grounding electrode, or the wire attached to the ground rod (grounding electrode conductor).  

There is usually a grounding jumper in the dryer that you can install with a three wire cord.  This provides protection in the event of a wiring fault without changing your wiring.  Unless you're 100% sure you've got a good ground, it might be better to just set up the dryer for a three wire power connection.