We discuss production of far-forward charm and anticharm quarks, $D$ mesons and antimesons, and neutrinos and antineutrinos from their semileptonic decays in proton-proton collisions at the LHC energies. We include the gluon-gluon fusion $gg\to c\overline{c}$, the intrinsic charm (IC) $gc\to gc$ as well as the recombination $gq\to Dc$ partonic mechanisms. The calculations are performed within the $k_T$-factorization approach and the hybrid model using different unintegrated parton distribution functions (uPDFs) for gluons from the literature, as well as within the collinear approach. We compare our results to the LHCb data for forward $D^0$-meson production at $\sqrt{s}=13\mathrm{TeV}$ for different rapidity bins in the interval $2<y<4.5$. A good description is achieved for the Martin-Ryskin-Watt (MRW) uPDF. We also show results for the Kutak-Sapeta (KS) gluon uPDF, both in the linear form and including nonlinear effects. The nonlinear effects play a role only at very small transverse momenta of $D^0$ or ${\overline{D}}^0$ mesons. The IC and recombination models are negligible at the LHCb kinematics. Both the mechanisms start to be crucial at larger rapidities and dominate over the standard charm production mechanisms. At high energies, there are so far no experiments probing this region. We present uncertainty bands for the both mechanisms. Decreased uncertainty bands will be available soon from fixed-target charm experiments in $pA$ collisions. We present also energy distributions for forward electron, muon, and tau neutrinos to be measured at the LHC by the currently operating $\mathrm{FASER}\nu$ experiment, as well as by future experiments like $\mathrm{FASER}\nu 2$ or FLArE, proposed very recently by the Forward Physics Facility project. Again components of different mechanisms are shown separately. For all kinds of neutrinos (electron, muon, tau), the subleading contributions, i.e., the IC and/or the recombination, dominate over light meson (pion, kaon) and the standard charm production contribution driven by fusion of gluons for neutrino energies $E_{\nu }\gtrsim 300\mathrm{GeV}$. For electron and muon neutrinos, both the mechanisms lead to similar production rates, and their separation seems rather impossible. On the other hand, for ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ neutrino flux, the recombination is further reduced making the measurement of the IC contribution very attractive.