The mechanical and thermal properties of a hybrid nanotube consisting of a coaxial carbon nanotube (CNT) inside a graphyne nanotube (GNT), i.e., CNT@GNT, are investigated in this paper by using molecular dynamics simulations. The results show that the mechanical properties of CNT@GNT under uniaxial tension depend on the nanotube chirality of its components. Specifically, the Young's modulus of the CNT@GNT structure with an inner zigzag CNT is larger than that of its counterpart with an armchair CNT, while CNT@GNT with an armchair CNT and a zigzag GNT is found to possess the largest tensile strength and fracture strain. In addition, a unique fracture behavior of the successive rupture of its two components is observed in CNT@GNT. The thermal conductivity of CNT@GNT is found to be almost independent of the nanotube chirality of its components but increases as the length and diameter of the CNT@GNT increase. Moreover, strain engineering is shown as an effective avenue to modulate the thermal conductivity of CNT@GNT, which can be enhanced by tension but reduced by compression. The analysis of the phonon spectrum and spectral energy density demonstrates that this strain effect originates from changes of the phonon group velocity and phonon scattering in the strained CNT@GNT.