These embeddings can indeed be parameterized. Explaining parameterizations of projective embeddings of a fixed degree is one of the main themes of the book: Geometry of Algebraic Curves: Volume I by Arbarello, Cornalba, Griffiths and Harris (ACGH). You should especially read chapter IV: The Varieties of Special Linear Series on a Curve. Although the exposition above is for smooth curves over $\mathbb C$ the main approach generalizes. All that one needs is that the $Pic$ functor is representable (At least if you want to parameterize by schemes and not by stacks).

The main approach of these kind of questions is as follows. If $g : C \to \mathbb P^n$ is a morphism then the pullback of the twisting sheaf $\mathcal O(1)$ on $\mathbb P^n$ gives a line bundle of degree equal to $e$. And if we write $\mathbb P^n = {\rm Proj \,} k[x_0,\ldots,x_n]$ then the $f^*(x_0),\ldots, f^*(x_n)$ are global section which will generate $f^*(\mathcal O(1))$. In fact from the linebundle $f^*(O(1))$ and the $n+1$ sections that generate it one can also contsruct the morphism $g$ again. So you see that your question is basically comes down to parameterizing linebundles together with a set of $n+1$ sections that generate it.

In particular the parameter space you are looking for will be an open subscheme of the scheme $G_e^n(C)$ in chapter IV of ACGH, it will not be all of $G_e^n(C)$ because you have to remove the closed part where the sections of de $g_e^n$ don't generate it.

These embeddings can indeed be parameterized. Explaining parameterizations of projective embeddings of a fixed degree is one of the main themes of the book: Geometry of Algebraic Curves: Volume I by Arbarello, Cornalba, Griffiths and Harris (ACGH). 

You should especially read chapter IV: The Varieties of Special Linear Series on a Curve, where they describe the parameterization. Although the exposition in this chapter is for smooth curves over $\mathbb C$ the main approach generalizes. All that one needs is that the $Pic$ functor is representable.

Some results about "how many ways", i.e. the dimension of the parameterization can be found in chapter $V$. I don't know how these results generalize to non reduced or reducible curves.

The main approach of these kind of questions is as follows. If $g : C \to \mathbb P^n$ is a morphism then the pullback of the twisting sheaf $\mathcal O(1)$ on $\mathbb P^n$ gives a line bundle of degree equal to $e$. And if we write $\mathbb P^n = {\rm Proj \,} k[x_0,\ldots,x_n]$ then the $f^*(x_0),\ldots, f^*(x_n)$ are global section which will generate $f^*(\mathcal O(1))$. In fact from the linebundle $f^*(O(1))$ and the $n+1$ sections that generate it one can also contsruct the morphism $g$ again. So you see that your question is basically comes down to parameterizing linebundles together with a set of $n+1$ sections that generate it.

In particular the parameter space you are looking for will be an open subscheme of the scheme $G_e^n(C)$ in chapter IV of ACGH, it will not be all of $G_e^n(C)$ because you have to remove the closed part where the sections of de $g_e^n$ don't generate it.