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Under interpolation-type assumptions such as the strong growth condition, stochastic optimization methods can attain convergence rates comparable to full-batch methods, but their performance, particularly for SGD, remains highly sensitive to step-size selection. To address this issue, we propose a unified stochastic trust-region framework that eliminates manual step-size tuning and extends naturally to equality-constrained problems. For unconstrained optimization, we develop a first-order stochastic trust-region algorithm and show that, under the strong growth condition, it achieves an iteration and stochastic first-order oracle complexity of $O(\varepsilon^{-2} \log(1/\varepsilon))$ for finding an $\varepsilon$-stationary point. For equality-constrained problems, we introduce a quadratic-penalty-based stochastic trust-region method with penalty parameter $\mu$, and establish an iteration and oracle complexity of $O(\varepsilon^{-4} \log(1/\varepsilon))$ to reach an $\varepsilon$-stationary point of the penalized problem, corresponding to an $O(\varepsilon)$-approximate KKT point of the original constrained problem. Numerical experiments on deep neural network training and orthogonally constrained subspace fitting demonstrate that the proposed methods achieve performance comparable to well-tuned stochastic baselines, while exhibiting stable optimization behavior and effectively handling hard constraints without manual learning-rate scheduling.