We present a theoretical study of the effect of electron-electron interactions and Sr doping on the electronic structure of infinite-layer (Nd,Sr)NiO$_2$ using the density functional+dynamical mean-field theory approach. In particular, we explore the impact of epitaxial compressive strain that experience (Nd,Sr)NiO$_2$ films on the electronic properties, magnetic correlations, and exchange couplings. Our results reveal the crucial importance of orbital-dependent correlation effects in the Ni $3d$ shell of Sr-doped NdNiO$_2$. Upon doping with Sr, it undergoes a Lifshitz transition which is accompanied by a reconstruction of magnetic correlations: For Sr $x<0.2$ (Nd,Sr)NiO$_2$ adopts the Neel $(111)$ antiferromagnetic (AFM) order, while for $x>0.2$ the $C$-type $(110)$ AFM sets in the unstrained (Nd,Sr)NiO$_2$, with a highly frustrated region at $x simeq 0.2$, all within DFT+DMFT at $T=290$ K. Our results for the Neel AFM at Sr $x=0$ suggest that AFM NdNiO$_2$ appears at the verge of a Mott-Hubbard transition, providing a plausible explanation for the experimentally observed weakly insulating behavior of NdNiO$_2$ for Sr $x<0.1$. We observe that the Lifshitz transition makes a change of the band structure character from electron- to hole-like with Sr $x$, in agreement with recent experiments. Our results for magnetic couplings demonstrate an unanticipated frustration of the Ni $3d$ magnetic moments, which suppresses magnetic order near Sr $x=0.2$. We find that the effect of frustration is maximal for Sr doping $x simeq 0.1-0.2$ that nearly corresponds to the experimentally observed doping value. We conclude that the in-plane strain adjusts a bandwidth of the Ni $x^2-y^2$ band, i.e., controls the effect of electron correlations in the Ni $x^2-y^2$ orbitals. The electronic properties of (Nd,Sr)NiO$_2$ reveal an anomalous sensitivity upon a change of the crystal structure parameters.
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