Frequency response of a (rectangular) integrator
Those are good questions! Here's my "two cents." Hilmar is correct, the integrator is not stable because there's a pole on the z-plane's unit circle. But I'll bet a pint of beer that the region of convergence is outside the pole (and therefore outside the unit circle) because the integrator's impulse response is causal.

Answer to question# 1: Matlab's freqz() command computes the freq response at zero Hz by evaluating a ratio of polynomials at various frequencies. And that ratio has a zero-valued denominator at zero Hz. So freqz() gives a freq mag response of infinity at zero Hz.  The DFT of an impulse response that's a finite-length sequence of ones, evaluated at zero Hz, is merely the sum of the impulse response samples.  The two approaches result in freq magnitude responses that are zero-valued at all frequencies except at zero Hz.  Both approaches give similar results.

Answer to question# 2: You could, if you wish, call an integrator a "filter." But it's unstable, and unstable filters are of no value in trying to filter real-world signals.