Transformers and switching converters
Firstly, dc current don't induce voltage in the secondary windings of
  any transformer.


Correct, a steady DC current doesn't induce secondary voltage. However, if the DC current is ramping up or ramping down in value it does.


  But in a flyback converter, I read that when the switch is ON, current
  will build up in the primary winding


Correct, and that ramping current induces a voltage. However, because of the polarity of the diode in the secondary circuit, that voltage can do no work i.e. it can't force a current into the load. So, current ramps up in the primary accumulating magnetic energy along the way and no energy transfers to the output load.


  prevent current flowing in the secondary winding, until the switch is
  open or OFF


Correct, when the switch turns off or goes open circuit, the magnetic energy forces current to flow into either the primary or the secondary. Given that the primary is now an open circuit, it forces current to flow into the diode and deliver the stored energy to the load. It can do this because the current is ramping down and that produces a reversal in voltage on the secondary that then forward biases the diode.


  When the transistor turns ON, current will be induced in the auxillary
  winding.


Current is not induced in magnetic elements; voltage is induced and, if there is a path for current to flow then current will flow. 


  So can a dc or a short constant current induce voltage in another
  winding of a transformer?


In the long term DC is static and doesn't have the credentials to induce voltage: -

$$\text{induced voltage} = \text{inductance}\cdot \dfrac{\text{change in current}}{\text{change in time}} \Longrightarrow V = L\cdot\dfrac{di}{dt}$$


  My main difficulty is, why are these voltages induced in the secondary
  winding when the input pulse is still ON(no change in current)?


When the primary is attached to the DC voltage source (V), the current rises linearly at a rate of V/L amps per second where L is the inductance of the primary winding. It's basically a re-arrangement of the formula I gave above. That linear rising current induces a voltage on the secondary terminals.