Measurements of integrated and differential cross-sections for W$^{±}$Z production in proton-proton collisions are presented. The data collected by the ATLAS detector at the Large Hadron Collider from 2015 to 2018 at a centre-of-mass energy of $ \sqrt{s}=13 $ TeV are used, corresponding to an integrated luminosity of 140 fb$^{−1}$. The W$^{±}$Z candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons or muons. The integrated cross-section per lepton flavour for the production of W$^{±}$Z is measured in the detector fiducial region with a relative precision of 4%. The measured value is compared with the Standard Model prediction at a precision of up to next-to-next-to-leading-order in QCD and next-to-leading-order in electroweak. Cross-sections for W$^{+}$Z and W$^{−}$Z production and their ratio are presented. The W$^{±}$Z production is also measured differentially as functions of various kinematic variables, including new observables sensitive to CP-violation effects. All measurements are compared with state-of-the-art Standard Model predictions from fixed-order calculations or Monte Carlo generators based on next-to-leading-order matrix elements interfaced with parton showers. An effective field theory interpretation of the measurements is performed, considering both CP-conserving and CP-violating dimension-6 operators modifying the W$^{±}$Z production. In the absence of observed deviations from the Standard Model, limits on CP-conserving Wilson coefficients are extracted using the transverse mass of the W$^{±}$Z system. For CP-violating coefficients a machine learning approach is used to construct an observable with enhanced sensitivity to CP-violation effects.