How far can we trust thermodynamic results at the critical point?
If the system is in equilibrium one can fully trust thermodynamics, even at the critical point. The derivation of the main thermodynamic formulas (e.g., in Chapter 9 of my online book) does not refer to the critical point, hence are universally valid. The singularities at the critical point appear only in the response functions (i.e., the behavior under systematic small changes), which poses no conceptual problem. The theoretically predicted asymptotic power laws at the critical point have been quantitatively verified by experiment.
But as one approaches the critical point, measurements get more and more difficult and hence more inaccurate, due to the increasingly long range fluctuations. On the other hand, the thermodynamic models used in practice for real substances usually have an inadequate (often mean field) analytic form to capture the correct singularity structure and hence are inaccurate, too.