If the check engine light is not coming on, there are probably no service codes in computer memory to read.
You can test for a failed catalytic converter without dropping the exhaust pipe; that is rather difficult on many vehicles.
If you have a vacuum gauge, attach it to the intake manifold and watch teh vacuum reading as you accelerate the engine; a restricted exhaust system will result in the vacuum dropping to or alose to zero, at which point the engine may stall. If the engine cannot pump exhaust out, it cannot draw enough air in and will be very low on power under load although it will usually idle OK. Another quick easy test is to temporarily remove the oxygen sensor that is screwed into the exhaust pipe ahead of the catalytic converter. ALthough the check engine light will come on and the engine may not run terribly well due to the lost oxygen sensor signal, if there is an exhaust restriction it will be partially relieved by the opening created where the sensor was, and will result in increased power and RPM range. If this is the case, you do need to remove the CAT to take a look inside.
Also, if you shake the exhaust system and can hear any chunks of debris rattling around inside, it is probably pieces of broken ceramic from a failed CAT.
There are other possibilities for what you describe though: the one that comes to mind first would be a jumped timing belt. If the cam and crank are slightly out of time due to a belt that has jumped one tooth due to belt stretch or worn tensioner components, it can result in symptoms similiar to what you describe. Checking cam timing is fairly easy on most engines; you line up the timing marks on the crankshaft and remove the upper belt cover(s) to make sure that the cam marks are also aligned.
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