Body-centered cubic (bcc) refractory multicomponent alloys are of great interest due to their remarkable strength at high temperatures. Optimizing the chemical compositions of these alloys to achieve a combination of high strength and room-temperature ductility remains challenging. Systematic predictions of these correlated properties across a vast compositional space would speed the alloy discover process. In the present work, we performed first-principles calculations with the special quasi-random structure (SQS) method to predict the unstable stacking fault energy (γusf) of the (11¯0)[111] slip system and the (11¯0)-plane surface energy (γsurf) for 106 individual binary, ternary and quaternary bcc solid-solution alloys with constituent elements among Ti, Zr, Hf, V, Nb, Ta, Mo, W, Re and Ru. Moreover, with the first-principles data and a set of physics-informed descriptors, we developed surrogate models based on statistical regression to accurately and efficiently predict γusf and γsurf for refractory multicomponent alloys in the 10-element compositional space. Building upon binary and ternary data, the surrogate models show outstanding predictive capability in the high-order multicomponent systems. The ratio between γsurf and γusf can be used to populate a model of intrinsic ductility based on the Rice model of crack-tip deformation. Therefore, using the surrogate models, we performed a systematic screening of γusf, γsurf and their ratio over 112,378 alloy compositions to search for alloy candidates that may have enhanced strength-ductility synergies. Search results were also validated by additional first-principles calculations.