Let $(M,g)$ and $(N,h)$ be two complete Riemannian manifolds and $f:M\rightarrow N$ be a differmorphism. If $\nabla$ is the Levi-civita connection of $M$ then is it true that the push-forward of $\nabla$ $f_*\nabla$ is the Levi-civita connection of $N$? If this statement true then is it possible that $F\circ \gamma$ is a geodesic in $N$ with connection $F_*\nabla$ where $\gamma$ is a geodesic in $M$?
No, you can generate many different connections $f_* \nabla$ by choosing different $f$, while the Levi-Civita connection is unique. The $f$ that this does hold for are known as affine transformations, which include isometries but in some cases other maps too - consider $\mathbb R^n$, where dilation and shear mappings preserve the connection but not the metric.
In the case that $f$ is affine, then yes, it sends geodesics to geodesics.