In this work, we proposed an autonomous Leslie-Gower model incorporating some biological factors such as fear and its carry-over effects, prey refuge, and nonlinear predator harvesting. In the proposed model, first, we examined the wellposedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in the intrinsic growth rate of a predator. Further, we computed and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf-bifurcation and direction and stability of the limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of the fear factor, carry-over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas, for small values of prey birth rate and harvesting effort, the system remains stable. However, when the intrinsic growth rate of a predator increases from a low value to a higher value, the system shows the transition from stability to instability and back to stability.