Is it possible to induce few volts IMPULS over 10 m distance?
Is it possible to induce for example 3 V short impuls on the secondary
  inductor when we press button?


Hopefully this should help: -



It's all about dumping as much current as you can into the transmit coil and, with your circuit you probably want to rethink the 1000 mF capacitor on the transmit side; it will have significant ESR and restrict current significantly.

It might be better if you concentrated on making both coils resonant at the same frequency - not only does this vastly increase the current into the transmit coil but it makes the receive coil more sensitive by the Q-factor.

So, will you get 3 volts with a 10 metre gap - not you won't easily - just do the math following my equation and you will see that it's likely you will need hundreds (if not thousands) of amps flowing in the transmit coil to achieve volts at the receive coil. Much beyond the transmit coil the flux density falls as a cube law with distance.

However, you might receive 3 mV that you can amplify up to 3 volts.


  If we have instead of inductor, resonat circuit, is it practically
  possible that signal last for a few seconds?


The better the tuning (higher Q) the longer the signal will last. 

The higher the frequency the higher the induced voltage will be because of rate of change of flux BUT the harder it is to dump the intial current into the transmit coil due to reactance. I would recommend wiring you transmit coil from high quality litz wire and making the diameter as large as you can.