Can a document contain its own hash?
It actually depends on what you consider a hash function.

If we take a CRC (Cyclic Redundancy Check) as our hash function, then it is quite feasible; a CRC has a property that every output bit is a linear (affine if the initial state is nonzero, but the difference is unimportant) function of the input bits; hence we can generate a set of linear equations over the various bits of X in the expression $CRC( Message || X || MoreMessage ) = X$, and attempt to solve that (and find the solutions, if any).

I would further note that CRC does meet the criteria you gave; it can potentially have a large range (just make the CRC state large enough), and at first glance, if you just look at inputs and outputs, it does appear random.

Of course, a CRC is not a secure hash function; however you didn't specify a secure hash function.