How to test digital filters?
Some thoughts on it at least.

First, if you can shown linearity and time-invariance and you know that the filter has the correct impulse response you are home. Given that the filter is stable. So in this case there is no need to run different other input signals and the frequency response is given based on the impulse response.

Checking impulse response is obviously quite simple. Linearity and time-invariance I guess is actually really complicated (any specific input signal you test may just work out of coincidence, but others may not). However, as the filter code in most cases shouldn't be that complicated, it should be possible to eye-ball the code and see that it does what is expected (no non-linear operators, no if-cases based on signal values etc).

Remains stability. For IIR filters this is a really complicated issue and I will not go into that discussion. For FIR filters, they will always be stable (unless they are implemented using some recursive algorithm). However, you may run into numerical issues relating to overflow (most likely not underflow) and round-off noise. You can find formal proofs that an FIR filter will not overflow (maximum output is the sum of the absolute values of the impulse response coefficients times the maximum input magnitude). For round-off errors you can find statistical expressions, but they are not of much use in formally proving anything (but rather as an argument that the expected output with "infinite precision" and the actual output still conforms).

Also not that all impulse responses etc should be evaluated using the actual (possibly rounded) coefficients not the (possibly higher precision) ones obtained from the filter design tool. This is because the filter will always realize the transfer function given by the coefficients that it use, no matter how they were derived. This may still not be the impulse response you get at the output though, as there may be round-off issues. An easy way to see that is to use a scaled impulse (i.e. not one) and note that typically not only the magnitude of the frequency-response will change, but also the shape. The filter will still implement the same transfer function, but the round-off noise added to the two different input signals are different.

Hope this gives you a bit more insight.