The force $T$ acts through the peg, therefore its moment about the peg is zero. Taking components of $T$ vertically (and horizontally) is not going to alter this situation.
You have actually already balanced moments when you assumed that the CG of the cone lies vertically below the peg. The only force acting on the cone which does not necessarily pass through the peg is the weight $W$ of the cone. When this force does pass through the peg then all moments on the cone are zero.
The cone can be balanced with various lengths of string. Ensuring that the CG lies below the peg constrains the relation between the angles $\theta_1, \theta_2$ which the string at the left and right makes with the horizontal. These angles are not necessarily equal. Making them equal is not required for static equilibrium. This is a geometrical constraint.
(a) Can be solved by applying geometry. Assume right hand upper edge is distance $x$ below the peg, then equate expressions for $\tan\theta$ in the right-triangles formed by right and left sections of the string.
(b) Can be solved by balancing forces vertically.