Do drugs always degrade after they have passed their expiration date?
Drugs do not have to degrade after their expiration date and passing the expiration date has more than one possible meaning. 

First of all chemical compounds vary in their susceptibility to breakdown over time.  If they are kept away from oxygen, high temperatures (or kept in very low temperatures), and they are of a robust chemical nature, they might be stored for much longer than their expiration date and remain just as strong. 

Over time, pharma compounds will chemically break down and lose their potency.  In many cases, you will find that they simply don't do anything when you take them.  This link cites a study that many drugs can retain their potency 15 years after they are made.  

There are important exceptions.  Two examples are tetracycline and acetaminophen (tylenol) whose breakdown products can increase the tendency for these drugs to induce liver damage.