Output length of an IIR Filter
To answer your last question, the impulse response having infinite length affects the output length in every possible way. In general, the output sequence of an IIR filter has an infinite length, even for a finite length input sequence. Of course, for stable filters and finite-length input sequences, the magnitude of the output samples decreases, and the output values become virtually zero after a while.

There are, however, specific input sequences that result in cancellations in the output, and, consequently, the corresponding output sequences have only a finite number of non-zero elements. A simple example is the first-order system

$$y[n]=ay[n-1]+x[n]\tag{1}$$

You can verify yourself that for the specific input sequence $x[n]=\delta[n]-a\delta[n-1]$, the output sequence is finite and simply equals $y[n]=\delta[n]$. However, note that the use of finite precision arithmetic in an actual implementation may still result in non-zero output samples, even in cases when ideally the output samples should cancel.