First, let's identify compound X by writing the hydrolysis reaction of diborane (B$_2$H$_6$).
Diborane reacts vigorously with water to produce boric acid and hydrogen gas.
B$_2$H$_6$(g) + 6H$_2$O(l) $\rightarrow$ 2H$_3$BO$_3$(aq) + 6H$_2$(g).
So, compound X is boric acid, H$_3$BO$_3$ or B(OH)$_3$.
Now let's evaluate the statements about boric acid.
I. It is a tribasic acid: Boric acid does not donate three protons. So, this statement is incorrect.
II. It is a weak monobasic acid: Boric acid is a weak acid. It does not act as a protonic acid (donating its own H$^+$). Instead, it acts as a Lewis acid, accepting a hydroxide ion (OH$^-$) from water.
B(OH)$_3$ + H$_2$O $\rightleftharpoons$ [B(OH)$_4$]$^-$ + H$^+$.
Since it releases one H$^+$ ion into the solution per molecule, it is considered a weak monobasic acid. This statement is correct.
III. It has a layer structure: In the solid state, B(OH)$_3$ molecules are linked by extensive hydrogen bonds, forming a two-dimensional planar layer structure. This statement is correct.
IV. It is highly soluble in water: Boric acid is only sparingly soluble in cold water, although its solubility increases with temperature. It is not considered highly soluble. This statement is incorrect.
The correct statements are II and III.