Quantum sensing and communication (QSC) is pivotal for developing next-generation networks with unprecedented performance. Many implementations of existing QSC systems employ Gaussian states as they can be easily realized using current technologies. However, Gaussian states lack non-classical properties necessary to unleash the full potential of QSC. This motivates the use of non-Gaussian states, which have non-classical properties beneficial for QSC. This paper establishes a theoretical foundation for QSC employing photon-varied Gaussian states (PVGSs). The PVGSs are non-Gaussian states that can be generated from Gaussian states using current technologies. First, we derive a closed-form expression for the generalized bilinear generating function of ordinary Hermite polynomials and show how it can be used to describe PVGSs. Then, we characterize PVGSs by deriving their Fock representation and their inner product. We also determine equivalence conditions for Gaussian states obtained from arbitrary permutations of rotation, displacement, and squeezing operators. Finally, we explore the use of PVGSs for QSC in several case studies.