  A reliable determination of the pole parameters and residues of nucleon
resonances is notoriously challenging, given the required analytic continuation
into the complex plane. We provide a comprehensive analysis of such resonance
parameters accessible with Roy-Steiner equations for pion-nucleon scattering -
a set of partial-wave dispersion relations that combines the constraints from
analyticity, unitarity, and crossing symmetry - most prominently of the
$\Delta(1232)$ resonance. Further, we study the Roper, $N(1440)$, resonance,
which lies beyond the strict domain of validity, in comparison to Pad\'e
approximants, comment on the role of subthreshold singularities in the
$S$-wave, and determine the residues of the $f_0(500)$, $\rho(770)$, and
$f_0(980)$ resonances in the $t$-channel process $\pi\pi\to\bar NN$. The latter
allows us to test - for the first time fully model independently in terms of
the respective residues - universality of the $\rho(770)$ couplings and the
Goldberger-Treiman relation expected if the scalars behaved as dilatons, in
both cases revealing large deviations from the narrow-resonance limit.
