Designs and simulations of 1.4, 1.5, 1.6, and 1.8-cell SRF gun cavities are studied at a linear accelerator gun system and they are compared with each other. We obtain high power beam from high energy electron beam produced at a photocathode with high quantum efficiency, high average current laser with high brightness, and high accelerating gradient. Beam energy of the whole accelerator system is effected mainly by the optimized RF power usage inside cells, the RF power interaction with particle beams, and the accelerated beam obtained by using SRF-gun cells. Thus, the optimized beam parameters, the RF parameters, and the accelerator gun cavity parameters that depend on each cavity geometries of SRF gun are obtained and presented in this paper. Additionally, the energy values of the electrons with optimized emittance at the gun exit are achieved and shown. We will conclude our paper by giving the gun output energies specifying which elliptical cavity-cell geometry is more appropriate to push the beam further in consistent conditions. For the cavity electromagnetic fields and the geometric designs of the cavities and the solenoid, we have utilized 2D Superfish/Poisson and 3D Computer Simulation Technology Programs. Also beam dynamics studies are included for only 1.6-cell SRF gun cavity which has the highest quality factor in order to figure out how the beam behaves along the beamline at the gun system. Transverse emittance (< 2 pi mm mrad) and beam energy (similar to 3.5 MeV) at the exit of the gun system are obtained as expected.