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Motivated by the recent discovery of superconductivity in the infinite-layer (Sr,Nd)NiO$_2$ films with Sr content $x \simeq0.2$ [Li et al., Nature (London) \textbf{572}, 624 (2019)], we examine the effects of electron correlations and Sr-doping on the electronic structure, Fermi surface topology, and magnetic correlations in (Nd,Sr)NiO$_2$ using a combination of dynamical mean-field theory of correlated electrons and band-structure methods. Our results reveal a remarkable orbital selective renormalization of the Ni $3d$ bands, with $m$*/$m\sim 3$ and 1.3 for the $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals, respectively, that suggests orbital-dependent localization of the Ni $3d$ states. We find that upon hole doping (Nd,Sr)NiO$_2$ undergoes a Lifshitz transition of the Fermi surface which is accompanied by a change of magnetic correlations from the three-dimensional (3D) Néel $G$-type (111) to the quasi-2D $C$-type (110). We show that magnetic interactions in (Nd,Sr)NiO$_2$ demonstrate an unanticipated frustration, which suppresses magnetic order, implying the importance of in-plane spin fluctuations to explain its superconductivity. Our results suggest that frustration is maximal for Sr-doping $x \simeq 0.1$--0.2, which is in agreement with an experimentally observed doping value Sr $x \simeq 0.2$ of superconducting (Nd,Sr)NiO$_2$.