Is zero-padding necessary for speech processing to satisfy linear convolution property?
My answer is not from speech processing angle but from a generic signal processing involving Linear and Circular Convolution.


  Is zero-padding necessary for speech processing to satisfy linear
  convolution property?


If the FFT size is $N$, and the length of result of linear convolution size is $\gt N$, then you need to do zero-padding. Otherwise it will result in time-aliasing in time domain. The effect of this is, at the output of FFT you will get an under-sampled representation of the theoretical continuous time DFT signal.
$$
(Y.*H)(e^{j2\pi k/n}) = (Y.*H)(e^{j\omega})|_{\omega = 2\pi k/N} 
$$
but
$$
(Y.*H)(e^{j\omega})|_{\omega = 2\pi k/N} \ne (\sum_0^{N-1}p[n]e^{-j\omega n})|_{\omega = 2\pi k/N}
$$
because of time aliasing. Here $p$ is the sequence obtained by linear convolution.


  What happens if I just stay with the circular convolution (i.e. NFFT
  length = signal length)?


Since your FFT length $N$ is equal to signal length $N$, the above time aliasing will happen.


  What if I just want to maintain the same signal length before and
  after processing?


Your FFT size has to be atleast equal to the resultant linear convolved signal length ($2N-1)$. Because that will be the length of signal you will be working with after the point-wise multiplication. So your FFT size has to be at least $2N-1$ before hand. See this question and its answers too Linear and Circular Convolution in Fourier Domain (DFT)