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We study sparse polynomials with bounded individual degree and the class of their factors. In particular we obtain the following algorithmic and structural results: 1) A deterministic polynomial-time algorithm for finding all the sparse divisors of a sparse polynomial with bounded individual degree. As part of this, we establish the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2) A poly(n,s^{dlog 𝓁})-time algorithm for recovering 𝓁 irreducible s-sparse polynomials of bounded individual degree d from blackbox access to their product (which is not necessarily sparse). This partially resolves a question posed in [Pranjal Dutta et al., 2024]. In particular, when 𝓁 = O(1), the algorithm runs in polynomial time. 3) Deterministic algorithms for factoring a product of s-sparse polynomials of bounded individual degree d from blackbox access. Over fields of characteristic zero or sufficiently large, the algorithm runs in poly(n,s^{d³log n})-time; over arbitrary fields it runs in poly(n,{(d²)!},s^{d⁵log n})-time. This improves upon the algorithm of [Bhargava et al., 2020], which runs in poly(n,s^{d⁷log n})-time and applies only to a single sparse polynomial of bounded individual degree. In the case where the input is a single sparse polynomial, we give an algorithm that runs in poly(n,s^{d²log n})-time. 4) Given blackbox access to a product of (not necessarily sparse or irreducible) factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm for finding all irreducible sparse multiquadratic factors of it (along with their multiplicities). This generalizes the algorithms of [Volkovich, 2015] and [Volkovich, 2017]. We also show how to decide whether such a product is a complete power (in case it is defined over a field of zero or large enough characteristic), extending the algorithm of [Bisht and Volkovich, 2025]. Our algorithms most naturally apply over fields of zero or sufficiently large characteristic. To handle arbitrary fields, we introduce the notion of primitive divisors for a class of polynomials, which may be of independent interest. This notion enables us to adapt ideas of [Bisht and Volkovich, 2025] and remove characteristic assumptions from most of our algorithms.