No need to take the cat off, just give it a thump with your hand, if it sounds like a baby's rattle then it's broken.
A decent tech or exhaust fitter will be able to tell if the exhaust is choked just by putting a hand over the tailpipe. If the pipe is clear the gas will 'pulse' as some air is drawn back in, if it's choked it's more of a steady blow.
If you want to get more technical you can remove the pre-cat lambda sensor and screw in a pressure gauge, or fit a vacuum gauge to the inlet manifold.
Not sure if a choked pipe would cause all your problems, but it's simple to check, if only for elimination purposes.
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