All optical spin noise spectroscopy is typically used to extract the virtually undisturbed spin dynamics from measurements of spin fluctuations in thermal equilibrium. In this thesis, the method is applied to study spin fluctuations in single positively charged InGaAs quantum dots beyond thermal equilibrium conditions, in view of the spin-photon interface provided by the optical transition. Spin noise spectroscopy with a resonantly driven optical transition additionally reveals the spin dynamics in the optically excited state, enables the characterization of the optical transition, and unveils a charge occupation noise which is naturally inherent in optically driven semiconductor quantum dots. The spin noise measurements in combination with a theoretical model show that the average spin relaxation rate of the optically driven spin is a mixture of hole-spin relaxation in the ground state and electron-spin relaxation in the excited state. The electron-spin relaxation is found to be on the order of a few 10 MHz, and dominates the average spin relaxation rate under quasi-resonant excitation. The dependence of spin dynamics and noise power on the laser detuning is used to determine saturation intensity, line width, and inhomogeneous broadening of the optical transition. It is shown that the unfavorable inhomogeneous broadening due to charge fluctuations in the quantum dot environment becomes very small and can even be absent in high-quality samples with a very low quantum dot density. Beyond the spin relaxation, the spin noise under quasi-resonant driving furthermore contains an additional contribution which is assigned to the temporary escape of the resident hole in the quantum dot. The detailed analysis of this occupation noise shows that the hole escape is initiated by non-radiative Auger recombination. The subsequent reoccupation of the quantum dot depends crucially on the solid-state environment. The intrinsic Auger rate is determined to be about 2 to 3 MHz for holes in InGaAs quantum dots. The reoccupation of the quantum dot by a hole is in general found to be slow in the investigated weakly p-doped sample, i.e., on a timescale of a few microseconds. In particular, it is shown that the presence of an ionized acceptor in the close vicinity of the quantum dot can significantly prolong the reoccupation time to several tens of microseconds as the result of a hole-capture competition between ionized acceptor and quantum dot.