The odd/even thing is exactly as expected. An even number of xor's would work only if the CRC had a zero initial value and zero final exclusive-or.

What does "Many checksum results of simple input tests (e.g. a small number of null bytes) also appear online in tables of CRC remainders." mean? Can you give an example?

Yes, for a 32-bit CRC it should be straightforward to determine the parameters with enough examples. If it really is a CRC. If you would like to provide examples here, we can see if it is.

In that case, you can start with, say, 20 examples of random 4-byte messages and their CRCs, in hex. That should be enough to determine a) if it's a CRC, b) the polynomial, and c) the ordering of the bits and bytes. If that verifies it's a CRC, then a few five-byte-message examples would be all that's needed to determine the initial value and final exclusive-or.