Filters generate output even if they are not connected to any signal and does respond differently to different signal sources
I simulated your filter in LTspice. In transient analysis with the input open circuit it broke into oscillation after 40ms. With a short circuit it took 200ms, but with 50uV of noise added it only took 20ms. 

AC analysis of the first stage shows phase increasing with frequency when passing through the center frequency of the filter. This is a sign of potential instability.



Increasing R2 from 1k&ohm; to 1.05k&ohm; I got this:-



The phase now flips to negative but the amplitude peaks sharply at over +20dB, indicating that the filter is right on the edge of being unstable.

With 1.1k&ohm; I got:-



The response is now similar to the original design with 1k&ohm;, but the phase is going in the correct direction for stability. This is confirmed by transient analysis, which does not show any hint of oscillation.

Trying to get the sharpest filter response with 5% tolerance components is dangerous because a small variation could push it into instability. You should either use 1% tolerance resistors, or 'detune' the filter and accept a wider bandwidth. Alternatively you could make a passive LC filter using high Q inductors, which is guaranteed to be stable.


  even if I don't connect any input signal(the output of mic), the
  filters output a sinusoidal-ish waveforms around 400-500 mVpp at
  their(filter's) designed center frequency when the loud speaker is on.
  I even connected input to ground, but did not help. One thing I could
  not understand is why filters interact with sound when there is no
  physical(electrical) connection with the microphone?


The speaker circuit might be inducing noise into the ground, power supply, or signal wires, which is 'shocking' the filters into oscillation, or you might have a 'ground loop' which is only present when the speaker is turned on.  


  Also, when connected with microphone, the filters still does not
  behave as before. I think it might be resulting from some kind of
  loading effect. There is a clear difference between mic out and signal
  generator out, which I could not figure out.


The microphone has a higher impedance than the generator. If the filters are close to instability then a small change (such as higher source impedance which effectively makes R1 larger) could push them over the edge. 

Adding a buffer between the microphone and filters is a good idea because it isolates them from the source and reduces interaction between them, but as you also have oscillation with the input short-circuited a buffer may not stop it. Even if it did, you should increase the filter stability margin to avoid problems in the future.