We present state-of-the-art string-based relativistic general-excitation-rank configuration interaction and coupled cluster calculations of the electron electric dipole moment, the nucleon electron scalar-pseudoscalar, and the magnetic hyperfine interaction constants (alpha(de), alpha(cs), A(parallel to), respectively) for the thallium atomic ground state P-2(1/2). Our present best values are alpha(de) = 558 +/- 28, alpha(cs) = 6.77 +/- 0.34 [10(-18)e cm], and A(parallel to) = 21172 +/- 1059 [MHz]. The central value of the latter constant agrees with the experimental result to within 0.7% and serves as a measurable probe of the P, T-violating interaction constants. Our findings lead to a significant reduction of the theoretical uncertainties for P, T-odd interaction constants for atomic thallium but not to stronger constraints on the electron electric dipole moment, d(e), or the nucleon electron scalar-pseudoscalar coupling constant, C-s.