Find filter coefficients to model a device using its measurement
We can represent a digital filter as:


A difference equation, relating input and output at every time step $n$:
$y[n] = \sum_{k=0}^{K-1} b[k] x[n-k] - \sum_{m=1}^{M} a[m] y[n-m]$. The filter coefficients are $b_k$ and $a_m$. In case of an FIR, $a[m] = 0\,\forall\, m $. This form is basically how the filter can be implemented (in the discrete time domain).
By z-Transform of the difference equation, we can also represent the above equation as: $Y(z) = \frac{\sum_{k=0}^{K-1} b[k] z^{-k}}{1 + \sum_{m=1}^{M} a[m] z^{-m}} X(z) = H(z)X(z)$. 
The frequency response is simply this z-Transform evaluated at the unit circle $H(z = \mathrm{e}^{j\omega})$ (this is often simply written as  $H(\omega)$).
The impulse response is the inverse transform of $H(\mathrm{e}^{j\omega})$. By convolution with the impulse response, we also can get the output of the filter, as in $y[n] = h[n] * x[n]$. However, $h[n]$ is only finite for an FIR (hence the name, obviously), where we can easily check that in fact it is simply the coefficients $b[k]$.


I hope this helps you with understanding the big picture. So to answer your questions:


The IFFT of the frequency response yields the impulse response. This corresponds directly to the filter coefficients only in the case of an FIR filter. In case you want to model your system as an IIR, you can use an algorithm such as the kind implemented in MATLAB's invfreqz, which is essentially an optimization to fit your frequency response to the complex rational function $H(z)$.
Yes, since $H(z)$ is a representation of the filter's difference equation, the coefficients of which are what we call filter coefficients. 


In any case, these are important basics you can learn from e.g. the classic book on DSP. Even the MATLAB documentation (which im sure you have already been checking) is a decent resource.