Proofs:

Statement (iv) follows from (ii). Assertion (iii) is a general fact about transitive group actions. To prove that the group action is transitive, let ${\tilde x}_1$ and ${\tilde x}_2$ be two points in the fiber $p^{-1}(x_0)$ and ${\tilde\gamma}$ be a path in ${\tilde X}$ joining ${\tilde x}_1$ and ${\tilde x}_2$. The image path $\gamma = p\circ {\tilde \gamma}$ is then a loop in $X$ based at $x_0$ and so represents an element of $\pi_1(X, x_0)$. Also ${\tilde\gamma}$ being the lift of $\gamma$ starting at ${\tilde x}_1$, we see that

$${\tilde x}_1\cdot [\gamma] = {\tilde \gamma}(1) = {\tilde x}_2.$$

Turning now to the proof of (ii), let ${\tilde x}_0\in p^{-1}(x_0)$ and $\gamma$ be an arbitrary loop in $X$ based at $x_0$. Then $[\gamma]$ belongs to the stabilizer of ${\tilde x}_0$ if and only if its lift starting at ${\tilde x}_0$ terminates at the same point ${\tilde x}_0$. That is if and only if $\gamma$ lifts as a loop based at ${\tilde x}_0$. But this is equivalent to saying $[{\tilde \gamma}] \in \pi_1({\tilde X}, {\tilde x}_0)$ and $\gamma = p_*[{\tilde \gamma}]$. Conversely if $[\gamma] \in \pi_1(X,x_0)$ is the image under $p_*$ of $[\tilde \gamma] \in \pi_1({\tilde X}, {\tilde x}_0)$ then ${\tilde\gamma}$ is a loop homotopic to a lift of $\gamma$ starting at ${\tilde x}_0$. But any two such lifts have the same terminal point which means

$${\tilde x}_0\cdot[\gamma] = {\tilde \gamma}(1) = {\tilde x}_0.$$

That is to say $[\gamma]$ belongs to the stabilizer of ${\tilde x}_0$ and that completes the proof.