4) This is a choice / assumption that is without bad consequences. And yes at any time, because if it is chosen as negatively charge, this system will never turn positively charged. 1) No, not yet. 2) The issue is that in some cases, especially this one, you have some constraints. The choice you wanted to take, necessarily disagrees with both KVL and Faraday's Law. The correct fix is so scrutinise the E fields by assuming a sensible charge distribution which fixes the E fields, and visualising

3) Why not assume that you added a tiny resistor, and then waiting for infinite time to pass? The oscillation will be damped away. The whole point is to invoke the idea from terminal velocity. I just set $V_{C_0} = V_C$, which is necessary because pure wires connect equipotentials.