When you're looking for AP Physics 1 problem ideas, start with old AP Physics B exams, especially the B exams between 1996 and 2014. These are a treasure of interesting situations, many of which can provoke more AP Physics 1 style questions than you can possibly ask in one school year.
In AP Physics 1, Coulomb's Law problems are restricted to the force exerted between two isolated charges. Usually Coulomb's Law questions will appear on the multiple choice. You'll see semi-quantitative questions, like "double the mass and halve the charge of one object, what happens to the electrical force?" You'll also see questions about the "relationship" between Coulomb's law and Newton's Law of Gravitation.
What about the free response? Interesting problems like 2010 AP B #3 are only appropriate for AP Physics 2 - that problem includes three charges, and asks about the vector sum of several forces.
Take a look, though, at 2009 AP B #2. There, two equal-length strings support objects of known mass and charge. The strings each hang an a known angle from the vertical. Ooh... just two charges, but the situation is ripe for questions that go beyond straight-up plugging into Coulomb's Law.
This problem isn't suited as-is for Physics 1, of course. It asks about electric field lines, which are not part of P1 (and are deprecated in P2 in favor of vector field notation). It asks about electric potential produced by several point charges. Skip those parts.
To rewrite for Physics 1, start with parts (c) and (d) of the original. Part (c) asked for a free body diagram of one of the two hanging objects. Part (d) brilliantly asked students to write two equations that could be solved simultaneously for the tension in and angle of the string. Explicitly, students were instructed NOT to solve the equations! Since AP Physics is not a math class, we don't care whether students can carry through the algorithmic solution to such a problem.* We do in fact care whether students can set up a system of equations, and then recognize whether the system is solvable.
* More precisely, we already know that our class of 20 students can create 20 different solutions.
So I assigned parts (c) and (d) only, and added a third part:
While the charge on object 1 remains positive Q, the amount of positive charge on object 2 is increased to 2Q. Describe any changes in the arrangement, and explain the physical reason for those changes.
I don't need the precise location of the new equilibrium position - that's way, way beyond the scope of AP Physics 1. All I'm looking to see here is indication that the separation between charges will increase, because the force of object 2 on object 1 (and vice-versa) is now larger. At the original locations, the horizontal component of the string's tension would no longer balance the electric force, and so there'd be an acceleration outward.
Another possible question based on this situation might be to ask whether, with charges Q and 2Q, the angles of the two strings would be the same as each other, or different from each other.
You got another question based on this situation? Add it in the comments!