We need the following theorem. It can be found in Munkres or deduced from Hatcher's proof that the fundamental group of $\pi_1(S^1) \cong \mathbb Z$.
Theorem 1: Lef $p : E \to B$ be a covering map; let $p(e_0) = b_0$. Let $f$ and $g$ be two paths in $B$ from $b_0$ to $b_1$; let $\tilde f$ and $\tilde g$ be their respective liftings to paths in $E$ beginning at $e_0$. If $f$ and $g$ are path homotopic, then $\tilde f$ and $\tilde g$ end at the same point of $E$ and are path homotopic.
Proof: See Munkres Theorem 54.3.
Lemma 2: Let $f$ be a loop in $X$. If $f$ is nullhomotopic, then there exists a nullhomotopy from $f$ which is a path homotopy.
Proof: Let $f : I \to X$ be a loop based at $x_0$. Let $a = h(1) = h(0)$. Suppose $f$ is nullhomotopic, then there exists a homotopy $H : I \times I \to X$ such that $H(s, 0) = f(s)$ and $H(s, 1) = e_c(s) = c$ where $c \in X$.
Define $\alpha_t : I \to X$ by $\alpha_t(s) = H(0, ts)$ and note that this is a path from $\alpha_t(0) = H(0, 0) = f(0) = a$ to $\alpha_t(1) = H(0, t)$.
Define $\beta : I \to X$ by $\beta_t(s) = H(1, t - ts)$ and note that this is a path from $\beta_t(0) = H(1, t)$ to $\beta_t(1) = H(1, 0) = f(1) = a$.
Define $h_t : I \to X$ by $h_t(s) = H(s, t)$ and note that this is a path from $h_t(0) = H(0, t)$ to $h_t(1) = H(1, t)$.
Hence, we've established that the following is well-defined: $\alpha_t * h_t * \beta_t$ is a loop based at $a$ for all $t \in I$. If we define $F : I \times I \to X$ by $F(s, t) = (\alpha_t * h_t * \beta_t)(s)$, then it is a path homotopy between $f$ and the constant map at $a$.
Lemma 3: Let $p : E \to B$ be a covering map. The lifting of a constant map is constant.
Proof: Let $f : I \to B$ be a constant path in $B$ and let $\tilde f$ be it's lifting from $B$ to $E$ via $p$. That is, $\tilde f : I \to E$ such that $p \circ \tilde f = f$.
Since $f$ is constant $\tilde f : I \to \pi^{-1}(c)$ where $f(s) = c$ for all $s \in I$. By property of covering maps, $\pi^{-1}(c)$ is discrete. Since $I$ is connected and $\pi^{-1}(c)$ is discrete, and a continuous map from a connected space to a discrete space must be constant, $\tilde f$ is constant.
Now we look at your $\star$ line:
Observe that $$\tilde h(1) = \tilde h \bigg( \frac12 + \frac12 \bigg) = \tilde h \bigg( \frac12 \bigg) + \frac{q}{2} = \tilde h \bigg(0 + \frac12 \bigg) + \frac{q}{2} = \tilde h(0) + q$$ and $$h(1) = h\bigg( \frac12 + \frac12 \bigg) = -h\bigg( \frac12\bigg) = -h\bigg(0 + \frac12\bigg) = h(0)$$ and so $h$ is a loop at basepoint $h(0) = h(1)$.
Suppose, by way of contradiction, that $h$ is nullhomotopic, then there exists a homotopy $H : I \times I \to S^1$ between $h$ and the constant map at $c$, $e_c$, for some $c \in S^1$.
By Lemma 1 we can suppose, without loss of generality, that $H$ is a path-homotopy in which case $c$ is the basepoint of $h$.
Since $e_c$ is constant, by Lemma 3 it's lifting, starting at $\tilde h(0)$, is constant.
Since $f$ and $e_c$ are path homotopic, by Theorem 1, $\tilde f$ and $\tilde e_c$ have the same end points which means that $\tilde h(0) = \tilde e_c(0) = \tilde e_c(1) = \tilde h(1)$ which implies that $q = 0$, a contradiction to $q$ odd. Conclude that $h$ is not nullhomotopic.
