Proof on page 215 of Miranda's book At the page 215, Miranda says that the dimension of the fiber of the map:
$$
\gamma: \{(X,D_{2g-1})\} \mapsto \{X_g\}
$$
where $\{(X,D_{2g-1})\}$ is the space of the pairs with $X$ an algebraic curve of genus $g$ and $D_{2g-1}$ is a divisor of degree $2g-1$ and $\{X_g\}$ is the space of algebraic curves of genus $g$, is equal to the dimension of the space of all divisors of degree $2g-1$, and this space has dimension $2g-1$. Why??
Moreover, at the same page, in exercise $F$ he asks to compute the number of parameters describing the space of the curves of degree $4$ in $\mathbb{P}^2$. 
Now, using the computation of the number of parameters for curves of genus $3$, I obtain the result. My question is: but the space $\{X_3\}$ also contains the curves with equation $y^2=h(x)$ with $\deg h(x)=7$ or $8$, so how can my computation give the expected result?

I think this is a tiny mistake in the book. With $D_{2g-1}$ Miranda means the space of effective divisors of degree $2g-1$, not all divisors of that degree. You can see that if you look at the preceeding page, where he says that the map $\{(X,D_{2g-1})\}\to \{(X,g^{g-1}_{2g-1})\}$ has fibers $(X,|D|)$ ($|D|$ parametrizes effective divisors linearly equivalent to $D$). Moreover, the space of all divisors of degree $2g-1$ obviously cannot have dimension $2g-1$, as the divisors of the form $p_1+\dots p_{2g}-q$ all have degree $2g-1$ and these divisors depend on $2g+1$ parameters. Now you should be able to see why fibers of $\gamma$ have dimension $2g-1$.
For the exercise F I think you are a bit confused about the flow of the arguments. First observation is that, all non-hyperelliptic smooth curves of genus 3 can be embedded in $\mathbb{P}^2$ as a smooth curve of degree 4 (Exercise:Why?). Then the exercise wants you to compute the parameters of degree 4 curves in $\mathbb{P}^2$. This is equal to: 
dimension of degree 4 polynomials in $\mathbb{P}^2$ - dimension of automorphisms of $\mathbb{P}^2$ $-1$. 
(The last -1 is due to projectivization.) If you compute this you will see that this number is equal to 6 and agrees with $3g-3$.
Now as you pointed out, our parameter count in $\mathbb{P}^2$ excludes hyperelliptic curves of genus 3, but it agrees with $3g-3$ anyway, because the locus of hyperelliptic curves inside $X_3$ is a subspace of lower dimension, in fact of dimension 5 (see p.213). I hope that helps.
