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This paper addresses a Hom-associative algebra built as a direct sum of a given Hom-associative algebra (A, ·, α) and its dual (A^∗, ∘, α^∗), endowed with a non-degenerate symmetric bilinear form B, where · and ∘ are the products defined on A and A^∗, respectively, and α and α^∗ stand for the corresponding algebra homomorphisms. Such a double construction, also called Hom-Frobenius algebra, is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to characterize the double construction of biHom-associative algebras, also called biHom-Frobenius algebra. Finally, a double construction of Hom-dendriform algebras, also called double construction of Connes cocycle or symplectic Hom-associative algebra, is performed. Besides, the concept of biHom-dendriform algebras is introduced and discussed. Their bimodules and matched pairs are also constructed, and related relevant properties.