Most large galaxies contain Super Massive Black Holes at their centers, drawing matter nearby to form swirling accretion disks emitting electromagnetic radiation. These are Active Galactic Nuclei. The brightest quasars are the most luminous Universe objects, impacting their host galaxy's morphology and formation. Studying their structure unveils galaxy evolution mechanisms. Recent accretion disk radius measurements challenge the standard model, prompting alternative models and methods. Observable at large distances, quasars serve as cosmological probes for the Hubble parameter, H0: Universe's expansion rate. Confirming tension between early and late-Universe H0 measures could suggest a new cosmological paradigm beyond flat-lcdm model. Gravitational lensing addresses both problems. Strong lensing occurs when massive objects, e.g., galaxies, between a quasar and observer create magnified images. Light's different paths due to varying gravitational wells cause time delays, providing information on H0, known as Time Delay Cosmography. Stars in the lensing galaxy can also magnify images via microlensing, probing quasar structure. The goal of this thesis is double: 1- using microlensing to study the inner structure of quasars, with a focus on one of the most interesting objects in this field and 2- measuring H0 using the second doubly-imaged quasar available so far to do TDC. More specifically, I have shown that the reverberation of radiation from the accretion disk by the surrounding Broad Line Region results in high-frequency features in the microlensing light curve that have been overlooked so far. Analyzing the microlensing light curves in the frequency domain has highlighted systematic when measuring accretion disk size when neglecting them. This novel technique also allowed me to accurately measure the size of the quasar's BLR, consistent with previous estimates. Furthermore, my work revealed that detecting unusually sustained periodic oscillations indicates that the quasar may host a Super Massive Binary Black Hole with a milli-parsec separation. The improbability of observing this kind of system right before they merge may indicate that their believed coalescence model is flawed. In the context of imminent wide-field surveys that will observe thousands of lensed quasars, I led the development of a neural network that can quickly and efficiently identify pairs of quasar images undergoing microlensing events. By highlighting the relevant time window for complementary observations, this forecasting algorithm will maximize the scientific output of such events. The TDCOSMO collaboration aims at measuring H0 with a 1% precision. As a contribution to this effort, I conducted the full analysis of the second doubly imaged quasar in the sample. This involved measuring the time delay, the lens velocity dispersion in 2D, analyzing the line-of-sight and modelling the lens mass. Each step required handling a different data set and investigating the measurement robustness. Despite the low number of images, the dimness of the lensed arc and the high number of perturbers, I could determine \hc with a precision of 12%. This measurement will contribute to improve the precision of the H0 measurement with the total TDCOSMO sample. Creating a sample of doubles for H0 measurement is crucial to identify potential selection biases present in the current sample dominated by quadruples and benefit from their large number to reach the 1% precision.