The recently discussed tendency of holes to generate nontrivial spin environments in the extended two-dimensional t-J model [G. Martins, R. Eder, and E. Dagotto, Phys. Rev. B 60, R3716 (1999)] is here investigated using computational techniques applied to ladders with several number of legs. This tendency is studied also with the help of analytic spin-polaron approaches directly in two dimensions. Our main result is that the presence of robust antiferromagnetic correlations between spins located at both sides of a hole either along the x or y axis, observed before numerically on square clusters, is also found using ladders, as well as applying techniques based on a string-basis expansion. This so-called "across-the-hole" nontrivial structure exists even in the two-leg spin-gapped ladder system, and leads to an effective reduction in dimensionality and spin-charge separation at short-distances, with a concomitant drastic reduction in the quasiparticle weight Z. In general, it appears that holes tend to induce one-dimensional-like spin arrangements to improve their mobility. Using ladders it is also shown that the very small J/t∼0.1 regime of the standard t-J model may be more realistic than anticipated in previous investigations, since such regime shares several properties with those found in the extended model at realistic couplings. Another goal of the present paper is to provide additional information on the recently discussed tendencies to stripe formation and spin incommensurability reported for the extended t-J model. These tendencies are illustrated with several examples.