This paper is organized as follows. In Section \[sec:notation\], we fix our notation for untwisted affine Lie algebras, and then recall some basic facts about semi-infinite LS paths, extremal weight modules, and their Demazure submodules. In Section \[sec:SMT\], we state a combinatorial version of standard monomial theory for level-zero extremal weight modules, and also its refinement for Demazure submodules; the proofs of these results are given in Sections \[sec:prf-SMT\], \[sec:prf-DC\], and \[sec:prf-Dem\]. In Section \[sec:SiSch\], we first review the formal power series model ${\mathbf{Q}_{G}}$ of semi-infinite flag manifold, and then introduce a semi-infinite version of Bott-Samelson-Demazure-Hansen tower for ${\mathbf{Q}_{G}}$. Then, we study the cohomology spaces of line bundles over ${\mathbf{Q}_{G}}$, and prove the higher cohomology vanishing; also, we describe the spaces of global sections in terms of Demazure submodules of extremal weight modules. As an application, we prove the normality of the semi-infinite Schubert varieties ${\mathbf{Q}_{G}}(x)$, $x \in W_{{\mathrm{af}}}^{\geq 0}$. In Section \[sec:K-SiFl\], after giving a definition of ${\widetilde{{\mathbf{I}}}}$-equivariant $K$-group $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ of ${\mathbf{Q}_{G}}$ (and $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ of ${\mathbf{Q}_{G}^{\mathrm{rat}}}$), we establish the Pieri-Chevalley formula (Theorem \[thm:PC\]) by combining our geometric results with the semi-infinite SMT. Also, in Section \[sec:nDAHA\], we show that our $K$-group $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ admits a natural nil-DAHA action. Appendices mainly 