You have two O2 sensors, one precatalyst and one after the first converter. Both of these sensors are critical to engine management, and will throw codes if they are out of specs. The second converter does not have a sensor, so engine controls doesn't care what shape it is in. That being said, there are State and Federal requirements regarding having these converters while operating on State or Interstate roads, and high population areas usually have annual inspections to ensure they are functioning properly, and your emissions are within specs.
Larger freer flowing mufflers, converters, and exhaust systems do seem to increase power slightly. About the best increase possible could be around 10% of your 125 hp. The stock hp is rated with all new and functioning equipment, so going beyond that level is hard to do with a motor with many miles on it.