Gyrotrons are a class of high-power vacuum-electronics microwave sources, which are envisioned to play an important role in the domain of magnetically confined fusion plasmas. Indeed, only gyrotrons are capable of producing continuous electromagnetic waves at sufficient power (>1 MW) and in the frequency range (~100 GHz) matching to one of the highest frequency collective modes of a magnetized plasma: electron cyclotron waves. The use of millimetre waves at this frequency has the added potential to very locally deposit energy (thus heating) and eventually control instabilities of a fusion plasma. The impressive advances in the gyrotron R&D makes electron cyclotron heating the reference and most promising auxiliary heating method in future fusion power plants. However, the gyrotron complexity is such that some gyrotron components still deserve some R&D. One of them is the Magnetron Injection Gun (MIG), where an annular electron beam is formed and accelerated in a region with externally applied electric and magnetic fields. For some gyrotrons, high-voltage arcing events, as well as significant leaking currents, have been observed in the MIG region and are believed to be associated to the formation of trapped electrons clouds, not caused by the main electron beam. These currents are disruptive and prevent the reliable operation of the device. Despite the very high-vacuum inside the tube, it is hypothesized that the clouds are formed by ionization of the residual neutral gas of the tube and are confined in regions where the presence of crossed externally applied electric and magnetic fields form a Penning-like trapping potential well. The clouds are annular nonneutral plasmas, that are in an exotic parameter regime where limited literature exists. This is due to their high electron density, strong externally driven azimuthal flow, non-negligible electron neutral collisions and formation without a controlled external source. The theoretical study of the formation, evolution and possible equilibrium of these clouds in the unexplored parameter range motivated the development of a new 2D (rz) particle-in-cell code, called FENNECS, capable of simulating the real MIG geometry, by using a novel, numerically efficient, finite element method based on weighted extended b-splines. This code also simulates the important electron-neutral collisions to reproduce the self-consistent formation and dynamics of the clouds. Furthermore, FENNECS is successfully verified and validated using experimental measurements obtained with a gyrotron suffering from problematic leaking currents. This represents the first self-consistent simulation of trapped electron clouds spontaneously forming in a realistic MIG geometry. The simulations show that not all potential wells are problematic, and support the relaxation of the "no potential-well" design criteria currently used in MIG design. In parallel, FENNECS simulations are used to derive a 0D analytical fluid model describing the clouds' quasi steady-state, which allows formulating scaling laws and increases our understanding of the clouds' dynamics. FENNECS is also used as a design tool for the new basic plasma physics experiment T-REX, whose goal is to study trapped electron clouds in configurations consistent with MIGs. Finally, a study of the diocotron instability in MIG configurations is presented, that shows the importance of this instability and opens the door to exploring it further in the context of gyrotrons.