If i use one ground wire for two outlets, will my equipment get damaged?
Generally, adding a ground wire is always better for electronic equipment.  It helps surge suppressors do their job by providing a ground reference.   It provides real chassis grounding which among other things gives a solid route for  static electricity to go to ground instead of getting inside equipment.  It also provides a fault path if any hot wires short to ground, which assures you get a breaker trip.  And it makes sure equipment chassis is not energized so it electrocutes someone. I'm glad you're doing it - it will help. 

That said, grounds will not cause cheap electronics with poor components  to suddenly become quality.  Grounds will not themselves "clean up" dirty, noisy AC power, although they will help surge suppressors do that.  Dirty power is often caused by your own equipment - for instance the air conditioner in your drawing.  You may want to protect the PC from the air conditioner.

Grounds also don't fix potential overloads, and you might have one here.  Fancy PC's aren't necessarily power hogs (the Mac Pro is only 2.5 amps) but "850 watt" PC power supplies are popular and they can draw 10 amps full bore (the difference being inefficiency).  A typical window air conditioner is 5-8 amps.  Loads which are continuous should only run at 80% of circuit capacity, that's 12A on a 15A breaker and 16A on a 20A breaker.  I would encourage you to investigate further to make sure those loads (and the other loads on that circuit) aren't too much.   To measure a device's actual current draw, gadgets like the Kill-A-Watt do nicely.