In geometric algebra, there are 2 complement operators (left complement and a right complement).
The formula you posted does not correspond to either of them. As described in my paper
The complement right is \star \langle\omega\rangle_p = \langle\tilde\omega\rangle_p I and the complement left is I\langle\tilde\omega\rangle_p
The metric is handled by the geometric product.
Yes, that’s correct modulo the grade-orientation and left/right handedness, it is orthogonal complement.
No, additional information is required, that article is not enough to define complement in GA (orientation).
Yes and No. In my own formulation of differential geometric algebra, there are 3 types of basis element that can square to the value of 0. Each of these 3 types has a different complement calculation involved. For example, there are Grassmann basis elements with metric 0, also null-basis elements from CGA, and finally I also have (what I would call) Leibniz-Taylor symmetric basis. Each is treated differently in my algebra.
What language are you using? Please share your code.