A clock. It runs all the time, it “sings” for hourly chimes, cuckoos or alarms, it doesn’t have a head (but does have hands) and you look a clock in the face to tell the time.
I am eight letters long and am kept a secret from everyone. Letters two through four spell an animal, letters four to the end make a weapon, letters one, two and eight make what you use in an exam and my third and fourth letters are the same.
A password. You should keep your password a secret from everyone, letters two through four spell ass, an animal also known as a donkey. Letters four to the end spell sword, which is a deadly weapon and the letters one, two and eight make pad, which is used for writing during an exam. Lastly, the third and fourth letters are both ‘s’