Troubleshooting a bent mech hanger
It looks like both the Hangar and the attachment from the rear mech to the hangar are bent?
You could try to straighten yourself with pliers or a small wrench but I would suggest a visit to the LBS who will use a Hangar Alignment Tool

http://www.parktool.com/product/derailleur-hanger-alignment-gauge-dag-2-2?category=Frame%20&amp;%20Fork

With a bent hangar you will never have smooth, crisp shifting at all.
That chain needs a degrease by the looks of the first photo. If it's an old chain, it may have "chain stretch" which will also affect shifting and start to ruin your cassette. Ask the LBS to check the chain length as well?