Does a helmet become useless after it slips from the hand and hits the ground?
Answering your general question:

PROBABLY not, according to Snell Foundation: 


  "I dropped my helmet! Do I have to go buy a new one?"
  
  Generally the answer is probably not.  Helmets are one-use items, but are quite durable otherwise, at least the ones we certify. Frequent dropping or spiking   a helmet on the ground, or other hard surfaces may eventually degrade the helmet's performance.  Similarly if the helmet falls to the ground at highway speeds unoccupied, the owner must be aware that some degradation may have occurred.   In general, the real damage comes when the helmet contacts an object with a head inside.  The Foundation recommends that if you are participating in an activity that requires that you wear a helmet, you avoid hitting stuff with your head.  It can be difficult to readily determine if a helmet has been damaged, and the protective capabilities compromised without a thorough inspection by a trained professional.  Some manufacturers may provide this service or direct you to these others that can perform these inspections.  The Foundation recommends that if you suspect your helmet may be compromised, then replace it. If the helmet has been involved in an impact while in use, replace it.  


But for your actual helmet you should have not any doubts - throw it away. As for motorcycle helmet it is very old. Even when new, your helmet presented worse performance than contemporary mid-end products.

Note for off-topic voters:
It is off-topic.

but

It is very important safety issue.