We derive new Hanson-Wright-type inequalities tailored to the quadratic forms of random vectors with sparse independent components. Specifically, we consider cases where the components of the random vector are sparse $α$-subexponential random variables with $α>0$. When $α=\infty$, these inequalities can be seen as quadratic generalizations of the classical Bernstein and Bennett inequalities for sparse bounded random vectors. To establish this quadratic generalization, we also develop new Bernstein-type and Bennett-type inequalities for linear forms of sparse $α$-subexponential random variables that go beyond the bounded case $(α=\infty)$. Our proof relies on a novel combinatorial method for estimating the moments of both random linear forms and quadratic forms. We present two key applications of these new sparse Hanson-Wright inequalities: (1) A local law and complete eigenvector delocalization for sparse $α$-subexponential Hermitian random matrices, generalizing the result of He et al. (2019) beyond sparse Bernoulli random matrices. To the best of our knowledge, this is the first local law and complete delocalization result for sparse $α$-subexponential random matrices down to the near-optimal sparsity $p\geq \frac{\mathrm{polylog}(n)}{n}$ when $α\in (0,2)$ as well as for unbounded sparse sub-gaussian random matrices down to the optimal sparsity $p\gtrsim \frac{\log n}{n}.$ (2) Concentration of the Euclidean norm for the linear transformation of a sparse $α$-subexponential random vector, improving on the results of G{ö}tze et al. (2021) for sparse sub-exponential random vectors.