Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$.
My approach: Suppose such field actually exist, consider a coordinate system $(U,x)$ on $M$. For hypothesis, the paths of $G$ ON $TU$ are given by $t\to(\gamma(t),\gamma'(t))$ where $\gamma$ is a geodesic, then I have a differential equation system. So, by uniqueness IF $G$ exist then is unique. This is right?? And the second question is, Always "any differential curve $\gamma(t)$ in $M$ determine a curve $t\to(\gamma(t),\gamma'(t))$ in his tangent bundle $TM$ "
For your second question, "yes", every smooth curve $\gamma$ in $M$ determines a unique lift $(\gamma, \gamma')$ in $TM$.
For your first question, you're on the right track: For each point $(x, v)$ in $TM$, there exists a unique maximal geodesic $\gamma$ in $M$ with $\gamma(0) = x$ and $\gamma'(0) = v$. The lift $(\gamma, \gamma')$ defines $G$ along a curve in $TM$ lying over $\gamma$. If $(x_{1}, v_{1})$ and $(x_{2}, v_{2})$ determine overlapping geodesics, the resulting definitions of $G$ agree where both are defined by uniqueness of geodesics, so $G$ is well-defined. Finally, every pair $(x, v)$ in $TM$ determines a geodesic locally, and so determines a curve in $TM$ along which $G$ is defined.