Prime/maximal ideals of $\mathbb{C}[x, y]$ containing a given ideal

The answer to your first question is "Yes", in fact if you consider $(p(x), g(y)) \cap \mathbb{C}[x]$, this must be a prime ideal (is an easy exercise) and then $p(x)$ must be irreducible. The same argument holds with $(p(x), g(y)) \cap \mathbb{C}[y]$.

Looking at the problem of count how many maximal ideals contains $I$, I think you can follow this way:

1. Call $M_i$ the maximal ideals you have found, then prove that $I=\cap M_i$

2. Suppose there's another maximal ideal $N$ such that $I \subset N$, then $\cap M_i=I=N \cap I = N \cap \bigcap M_i $. You can use now the following Lemma to conclude:

Lemma Let $P$ be a prime ideal. If $I_1 , \dots , I_n$ are ideals such that $\cap I_i \subseteq P$, then there's an index $j$ such that $I_j \subseteq P $. (Try to prove it!)

If I well remember there's another way to compute such number of maximal ideals, using a bit of Algebraic Geometry and Theory of Grobner Basis.

We starts from the observation that, in an algebrically closed field, a point of a variety $V(I)$ corresponds to a maximal ideal containig $I$. Follows that, if $V(I)$ is finite, it is contained in a finite number of maximai ideals, many as its points.

Now, we link tha finiteness of $V(I)$ to the Grobner Bases by the following result.

Theorem A variety $V(I)$ han a finite number of points iff there's only a finite number of monomials not contained in the Leading Terms Ideal of $I$.

Then the following lemma (It's a vague image in my memory : I hope there's no mistakes in its assert ) can easily solve you problem:

Lemma Let $k$ me an algebricaly closed field and $I$ an ideal of the ring $k[X_1, \dots, X_n]$ such that $V(I)$ is finite. The following integers are equal:

1. The number of points of $V(I)$.
2. The dimension of the ring $k[X_1, \dots, X_n]/I$ as $k$- vector space.
3. The number of monomials not contained in the ideal of the leading terms of $I$

It seems to be more complicated, but with this theorem is immediate, just calculating a Grobner Bases, to obtain a lot of information about the ideal $I$ and its variety.

In your case, for example, is very easy to check that $x^2-1$ and $y^3-1$ are a Grobner Bases of $I$ and that (for example) there's only 6 monomials not containden in the Leading Terms Ideal of I.

I'm conscious of the fatc that, if you're a student and don't know anything of this argumens, this is only an unintelligible speech, but I think is, in every case, very interesting.