Quantitative analysis the role of turbulence in magnetic self-generating and organization, plasma self-feeding and sustaining, which closely related to the evolution of neutral points and flux ropes, is the most important to understand magnetic energy release-conversion, plasma heating, charged particles energization and acceleration in 3D large temporal-spatial scale turbulent magnetic reconnection (LTSTMR; observed current sheets thickness to electron characterize length, electron Larmor radius for low- and electron inertial length for high-, ratio in 10E10 ~ 10E11 order; observed evolution time to electron gyroperiod ratio in 10E10 ~10E11 order) that cover from Earth's magnetosphere to solar eruption and other astrophysical phenomena. As the first part of this work, a relativistic hybrid particle-in-cell & lattice Boltzmann (HPIC-LBM) model that describes the continue kinetic-dynamic-hydro full coupled temporal-spatial scale LTSTMR evolution process. First, based on the governing equations for resistive relativistic Magnetohydrodynamics (MHD), the relativistic discrete distribution function for magnetic field (D3Q7), electric field (D3Q13), electromagnetic field (D3Q13), as well as charged particle (D3Q19) and neutral particle (D3Q27) of different species plasma are obtained for RHPIC-LBM lattice grid. Then, the RHPIC-LBM numerical process, the algorithm, the code flow chart as well as GPU-CPU heterogeneous frame work is described. Last, the present model  is tested and validated on Tianhe-2 from National Supercomputer Center in Guang Zhou (NSCC-GZ) with 10,000 ~ 100,000 CPU cores & 50 ~ 120 hours per case for investigating solar atmosphere activities LTSTMR from pico-scale  (10E-2 m ~10E5 m),  nano-scale (10E5 m ~10E6 m), micro-scale (10E6 m ~ 10E7m), macro-scale (10E7 m ~ 10E8 m) and large-scale (10E8 m ~10E9 m). All simulation results are consistent with observations. Based on the model and code developed in first part, we investigated the turbulence of helical magnetic structure, current density vector fields, magnetic fields self-generating and organization, plasma self-feeding and sustaining, ion and electron acceleration of 3D LTSTMR with 100,000 CPU cores on Tianhe-2 in the second part. By simulation evidences,  we discovered and confirmed following results:  i) Slipping magnetic reconnection (MR) exists in the adjacent magnetic field lines (MFLs) compress-stretch-slip process on the quasi separatrix layers (QSLs, drastic change MFL linkage span) and adjacent MFLs cut-break-reconnect MR exists on the separatrix surfaces (SLs) that are consistent to observations; ii) The slipping MR and the MFLs cut-break-reconnect MR is closed link with the oblique and resistive tearing instabilities, respectively. The 1st type MR forms O-type neutral points, and the 2nd type MR forms X-type neutral points. The magnetic energy conversion is dominated by turbulence induced oblique instabilities (O-type) in 3D model instead of resistive tearing instabilities (X-type) in 2D/2.5D model, which are very well consistent with 3D observations; iii) The magnetic energy conversion is produced in the interaction of plasmoid-to-flux rope, plasmoid-to-plasmoid, and flux ropes-to-flux ropes. The turbulent acceleration is an independent acceleration mechanism in LTSTMR, induced by the interaction of waves-to-waves and waves-to-particles, and is different from original hybrid acceleration mechanism (composed of parallel electric fields, betatron, shock and Fermi) ; iv) The particles can be energized and accelerated in a longer time scale (~30 s) and can be accelerated to relativistic energy bin after pre-accelerated by Fermi-Betatron-shock wave acceleration which are agreed with observations.