This article conducted an analysis on quantitative properties and stability in a $\beta$-normed space for a category of boundary value problems of nonlinear two-term fractional-order sequential differential equations with variable coefficients. The original problem was converted into an equivalent integral equation. Banach's fixed-point principle and Shaefer's fixed-point theorem were exploited to ensure that two existence conditions of the solutions for the problems were established. In addition, the stability known as $\beta$-Ulam-Hyers for such problems has also been analyzed. Illustrative examples demonstrated practical applications of the work.