We have endeavoured to understand the formation and evolution of the black hole (BH) X-ray binary LMC X-3. We estimated the properties of the system at four evolutionary stages: 1) at the zero-age main-sequence (ZAMS), 2) immediately before the supernova (SN) explosion of the primary, 3) immediately after the SN, and 4) at the moment when Roche-lobe overflow began. We used a hybrid approach that combined detailed calculations of the stellar structure and binary evolution with approximate population synthesis models. This allowed us to estimate potential natal and the evolution of the BH spin. We incorporated as model constraints the most up-to-date observational information throughout, which include the binary orbital properties, the companion star mass, effective temperature, surface gravity and radius, and the BH mass and spin. We find at 5% and 95% confidence, respectively, that LMC X-3 began as a ZAMS system with the mass of the primary star in the range M_1,ZAMS_=22-31M_{sun} and a secondary star of M_2,ZAMS=5.0-8.3M_{sun}, in a wide (P_ZAMS>~2.000days) and eccentric (e_ZAMS_>~0.18) orbit. Immediately before the SN, the primary had a mass of M_1,preSN_=11.1-18.0M_{sun}, but the secondary star was largely unaffected. The orbital period decreased to $0.6-1.7days and is still eccentric 0>=e_preSN>=0.44. We find that a symmetric SN explosion with no or small natal kicks (a few tens of km/s) imparted on the BH cannot be formally excluded, but large natal kicks in excess of >~120km/s increase the estimated formation rate by an order of magnitude. Following the SN, the system has a BH M_BH,postSN_=6.4-8.2M_{sun} and is set on an eccentric orbit. At the onset of the Roche-lobe overflow, the orbit is circular and has a period of P_RLO_=0.8-1.4days.