Chemistry - Writing half equation confusion

Can I write $\ce{X + 2 e- -> X^2-}?$ Why can't I start with $\ce{X}$ instead of $\ce{X^2+}?$

The question says that the metals are $\ce{A(s), B(s), C(s) and D(s)}$. That means the letters stand for the uncharged element, not for something more arbitrary. In the table, the letters are used again, in the following way (shown for A):

$$\ce{A(NO3)2(aq)}$$

Because nitrate has a negative charge, the metal A forms an ion in water with a positive 2 charge ($\ce{A^2+}$).

$$\ce{A(NO3)2(s) -> A^2+(aq) + 2NO3-(aq)}$$

With the same argument, you can figure out that B also forms a divalent cation in solution, while C and D are monovalent. There is no flexibility of having $\ce{A}$ stand for an ion, or for the formation of a metal anion; you should stick to the species and symbols in the text and table of the question.

Examples where letters stand for something else

In acid/base chemistry, A and B are used in the following way:

$$\ce{AH <=> A- + H+}$$

$$\ce{B + H+ <=> BH+}$$

If you look closely, $\ce{B}$ stands for a base (with conjugate acid $\ce{BH+}$) and - which might cause confusion - $\ce{AH}$ stands for an acid (with conjugate base $\ce{A-}$). Neither A nor B have to be a single atom, and their charge could be anything. However, $\ce{B}$ and $\ce{A-}$ have one charge less than $\ce{BH+}$ and $\ce{AH}$, respectively.

In redox (bio)chemistry, $\ce{NADH and NAD+}$ are used. Again, the charges are to be interpreted as relative, and the relationship between the two is given by:

$$\ce{NAD+ + H+ + 2e- <=> NADH}$$