#05AB1E, 13 bytes

2.ÆDí«ε`Å?}O_

Too long.. Initially I had a 9-byte solution, but it failed for the duplicated key test case.

Try it online or verify all test cases.

Explanation:

2.Æ             # Get all combinations of two elements from the (implicit) input-list
   Dí           # Duplicate and reverse each pair
     «          # Merge the lists of pairs together
      ε         # Map each pair to:
       `        #  Push both strings to the stack
        Å?      #  And check if the first starts with the second
          }O    # After the map: sum to count all truthy values
            _   # And convert it to truthy if it's 0 or falsey if it's any other integer
                # (which is output implicitly as result)

05AB1E, 13 bytes

2.ÆDí«ε`Å?}O_

Too long.. Initially I had a 9-byte solution, but it failed for the duplicated key test case.

Try it online or verify all test cases.

Explanation:

2.Æ             # Get all combinations of two elements from the (implicit) input-list
   Dí           # Duplicate and reverse each pair
     «          # Merge the lists of pairs together
      ε         # Map each pair to:
       `        #  Push both strings to the stack
        Å?      #  And check if the first starts with the second
          }O    # After the map: sum to count all truthy values
            _   # And convert it to truthy if it's 0 or falsey if it's any other integer
                # (which is output implicitly as result)