In this paper, the authors establish new versions of the Farkas lemma for systems which are convex with respect to a cone and convex with respect to an extended sub linear function under some Slater-type constraint qualification conditions and in the absence of lower semicontinuity and closedness assumptions on the functions and constrained sets. The results can be considered as counterparts of some of the earlier corresponding results in Dinh et al. (SIAM J. Optim. 678-701, 2014). As consequences, the authors get extensions of the HahnBanach theorem for extended sublinear functions (the situation where the celebrated Hahn-Banach theorem failed). The results obtained are then applied to provide duality results and optimality conditions for a class of composite problems involving sublinear-convex mappings. Two special cases are examined at the end of the paper. In the first one, the results give rise to some generalized Fenchel duality theorems while in. the second one, in normed spaces, the result leads to the separation theorem for convex sets (not necessarily closed) in normed spaces.