1. We don't say to counter-argue but to argue back.
2. We can say to argue over something or someone. It means to argue about such things as ownership (we argued over the dog - i.e. about who should own it), for a detail we'd more usually say argued about (we argued about whether or not the US should get out of Iraq). I can't think of an example where we'd say we argued on something or someone.
3. As Lilliput has shown we argue with someone about or over something.
One can 'talk over' somebody else if maybe that was also what you were looking for Michael_cycle, that would mean that instead of interrupting someone you simply start to talk at the same time they are talking - this type of bad communication would usually happen in a heated argument, which could be the situation you are referring to.