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We compare and examine the influence of Hom-associativity, involving a linear map twisting the associativity axiom, on fundamental aspects important in study of Hom-algebras and (σ,τ)-derivations satisfying a (σ,τ)-twisted Leibniz product rule in connection to Hom-algebra structures. As divisibility may be not transitive in general not necessarily associative algebras, we explore factorization properties of elements in Hom-associative algebras, specially related to zero divisors, and develop an α-deformed divisibility sequence, formulated in terms of linear operators. We explore effects of the twisting maps σ and τ on the whole space of twisted derivations, unfold some partial results on the structure of (σ,τ)-derivations on arbitrary algebras based on a pivot element related to σ and τ and examine how general an algebra can be while preserving certain well-known relations between (σ,τ)-derivations. Furthermore, new more general axioms of Hom-associativity, Hom-alternativity and Hom-flexibility modulo kernel of a derivation are introduced leading to new classes of Hom-algebras motivated by (σ,τ)-Leibniz rule over multiplicative maps σ and τ and study of twisted derivations in arbitrary algebras and their connections to Hom-algebra structures.