Do variables in a convex optimization problem have to satisfy the inequality constraints only at the optimum or throughout the iteration from initial point to convergence?
Mathematically, it doesn't matter but if those variables were variables of a physical system that has physical limitations, then they would always have to satisfy the constraint, be it optimal or transient.
It depends on the optimization algorithm you choose. There are "infeasible" methods, in which you only have convergence to a feasible point, and there are "feasible" methods, in which all the iterates are always feasible.
An example of a feasible method is gradient projection. An example of an infeasible method is dual proximal gradient.