Altermagnets combine compensated collinear magnetic order with momentum-dependent spin splitting. A key question is how such spin splitting can arise from electronic interactions in a spin-degenerate Dirac semimetal and what spin-resolved transport signatures it produces. We address this issue using a staggered $\pi$-flux Hubbard model. Within a homogeneous, collinear, two-sublattice self-consistent Hartree-Fock ansatz, we show that the Hubbard interaction drives the Dirac semimetal into an altermagnetic phase with an even-parity, $d_{xy}$-wave spin splitting protected by mirror-spin symmetry $[C_2||M_y]$. The interaction-generated order gaps the Dirac cones at half filling and gives rise to an altermagnetic metal upon doping. In the spin-orbit-free limit, we compute the second-order electric and thermoelectric responses and identify a clear channel dependence. The nonlinear charge Hall and Nernst responses are already permitted in the spin-degenerate semimetal and are further reshaped by the interaction-driven Fermi-surface reconstruction, making them nonunique indicators of altermagnetic order. By contrast, the nonlinear spin Hall and spin Nernst responses vanish identically in identical-spin-block parent and become finite in the pure-spin $yxx$ channel of the ordered state, accompanied by a characteristic sign reversal across half filling. The mirror-spin symmetry produces an orthogonal pure-charge $xyy$ channel and opposite domain parities. These results establish a symmetry-resolved two-channel nonlinear transport diagnostic based on the parent-state null and directional charge-spin separation.