Are large polynomials secure for order preserving "hashing"? (newbie question)
The basic method is easily cracked: it is well known how to find a polynomial of degree at most $k$ from $k+1$ (input, output) pairs; that's the polynomial interpolation problem. There are numerous ways to efficiently carry it for high degree and large integers (one such method is to carry it modulo some medium primes, and use the Chinese Remainder Theorem in the end).

"pre-transforming" alone does not help at all against the above.

"post-transforming" and "adding a random deviation" makes recovering the polynomial more difficult. However, some simple attacks remain feasible, depending on adversarial model. In particular, if the adversary can query the function for arbitrarily large input: the degree of the polynomial and the highest degree coefficient of the polynomial divided by commonDivisor are easily found; then the attack can be repeated by subtracting known high coefficients, until finding all coefficients; in this setup, commonDivisor does not help much.

I do not see how to make the idea secure in a cryptographic sense.

Also: It does not seem that the "post-transforming" method proposed is guaranteed to maintain the property that distinct input leads to distinct output.

Addition following comment: if input is not known, that makes attack harder; however


you proposed: lets assume I have transformed all integers from 0 to 1000, and in cryptographic terms that's to be assimilated to a lot of known plaintext;
known plaintext has been a standard assumption for about as long as cryptography matured to a science, for reasons similar to the above;
as the saying goes: attacks only get better; they never get worse.