I have a 1999 Chevy Blazer, the check engine light (p0420 - catalyst efficiency below threshold) came on about 6 months ago. A mechanic told me not to worry about it. This morning the engine starting missing and a new code appeared (p0303 - cylinder number 3 misfire). I replaced the plugs and wires but this did not fix the problem. Is this all due to the catalytic converter?

 

While it is possible for the catalytic converter to be at fault, you should run through the diagnosis. I would recommend finding another mechanic as your previous mechanic did nothing to help you out concerning his advice to ignore the P0420 code.

Here are the trouble code diagnostic charts:
P0420 - Catalyst System Efficiency Below Threshold (Bank 1)
P0300 - Random/Multiple Cylinder Misfire Detected

The P0300 code will go through most of the same steps as the cylinder specific codes.

How did the #3 plug look when you swapped it out?

Most exhaust shops have the equipment to run an exhaust back pressure test where they will install a pressure gauge before and after the catalytic converter (using the O2 sensor ports). If the pressure drop over the catalytic converter is too great, that would indicate a restricted catalytic converter that is in need of replacement.