I did not read all the posts, so I hope someone did not suggest it already.

Yes its possible. After the microphone you might need an amplifier. After the amplifier you need a band frequency filter for 10-11kHz, after the band filter the signal will be only 10-11KHz with some accuracy, that signal is transformed into a 0-1KHz signal with the same voltage and current amplitude.

 

For you initial description, detecting a tone, then gating another tone on
the prior suggestion LM567 by drc_567 is a natural.

If you wanted to detection and generation on one chip use a UP, detection by
measuring freq of input, generation using wave table and DAC. If fairly noisy
environment add some DSP or analog filter into the mix. There is UP out there
than can do all of this.

Regards, Dana.