Simulating the impulse response of a multirate digital filter
By definition, the Impulse Response of a Filter is the response the filter has to an Impulse: https://en.wikipedia.org/wiki/Impulse_response

Since your Cascaded Integrator Comb Filter is an optimized FIR filter, it's response will be Finite in length (as opposed to IIR - Infinite Impulse Response) and you can calculate this response by sending a single maximum value followed by zeros through it.  

You mentioned [1, 0, 0, 0] is not an impulse response since 0 represents -1.  I believe the value that represents 0 would be half of the unsigned maximum value in this situation.  If your values are Unsigned integers from 0 to 255 with 0 representing the largest negative value (-1) and 255 representing the highest positive value (+1), you should put a maximum value (255) followed by zero values (128).  After the filter, you will probably shift/scale/truncate back to the previous bit-width to account for bit growth/gain of the filter.  This will give you the values of your Filter's Impulse Response.  

You have a right idea for calculating the Impulse Response when you tried input '10101010...'  and '11010101010...' (these are really like putting in [1 -1 1 -1 ...] and [1 1 -1 1 -1 1 ...]).  Since FIR filters (CIC in this case) are linear systems, you can use the Superposition rule and add the inputs and then filter.  You will get the same answer as filtering the signals separately and then adding. That is why your experiment gives you twice the filter impulse response.  If you add after you filter, you will still get twice the filter impulse response.

I proved this to myself with this example (MATLAB):

If a is your filter:

a = [1 -1 1 1 1 -1]


and b and c are your inputs:

b = [1 -1 1 -1 1 -1 1 -1 1 -1 1]
c = [1 1 -1 1 -1 1 -1 1 -1 1 -1]

filter(a,1,b) + filter(a,1,c) = [2 -2 2 2 2 -2 0 0 0 0 0]


which is two times your filter impulse response.

However, you still get the filter impulse response by inputting a single max input:

filter(a,1,[1 0 0 0 0 0 0 0 0 0 0 ]) = [1 -1 1 1 1 -1 0 0 0 0 0]


For a very good treatment of CIC filter analysis, see:
http://dspguru.com/sites/dspguru/files/cic.pdf

To calculate the Frequency Magnitude Response of your filter, you can take the FFT of the coefficients.  To plot it, you should do something like this in MATLAB:

impResp = [1 -1 1 1 1 -1];

freqDomain = fft(impResp);
fSamp = 100;
frequencies = -fSamp/2:fSamp/(numel(impResp)-1):fSamp/2;

figure
plot(frequencies,db(abs(fftshift(freqDomain))))
xlabel('frequencies')
ylabel('db')


This filter does not have a very interesting Impulse response, though.