Is the solution that SQP produces always the best or unique? How can I seek for other solutions?
By varying the start point, $x_0$?
Also, any idea why the start point may be "absurdly" large compared to the solution. I'm trying e.g. $x0=[50000.0,1000.0,3000.0,5000.0]$. When the sol is $[1, 0, 0, 1]$ and at step 2. the algorithm is already around $[1,1,1,1]$.
Whether a solution is unique depends on the problem, not on the algorithm that you use. You should check if the optimization problem is strictly convex. If there are multiple optima, trying different starting solutions is a valid strategy.
If SQP can make much progress in just one iteration means that your objective function can be closely approximated by a convex quadratic function, at least in terms of the location of the optimum.