My car failed the state emissions test with high HC readings

Well I took my 2003 hyundai tiburon to have emissions testing done on it and it failed due to High NOx readings. It states here on this test sheet that High NOx readings are the result of high combustion temperatures and/or High combustion pressures. So My step dad had suggested we had it looked at and the mechanic said that the catalytic converter and O2 sensor need to be replaced. Also something about the spark plugs and he was going to take a look at the vacuum hoses. But overall I was seeking a second opinion as to what you guys think my problem might be not passing the NOx GPM part of the test. Also my check engine light is on (the girl I bought it from said it turned on when they put a new header on) not sure if that was the truth but anyways, it also idles wierd sometimes when I start it up, it will start fine and then it idles up and down so I give it a little rev and its good it does that periodically at stop lights as well. Oh and I cant forget the girl I bought it from also mentioned when they put in the high performance exhaust they had removed the O2 sensor, but yes on the emissions test it had passed so I thought it was wierd but I have no clue. Well I hope you guys have some tips for me and I do really appreciate you guys taking the time to help me out!

Thanks,
Seth

Ok that reminds me the mechanic said that when he took a look at the catalytic converter he said they had gutted it out, and im not sure what you mean by code numbers for the failure.

The reading numbers? (3.0639) Limit: 1.5000, I live in Colorado.

Or I have these other numbers on the other side it says 0420, catalyst system efficiency below threshold (bank 1)
and 0107, MAP/Barometric Pressure circuit low input.

0402: Catalyst system efficiency below threshold.
0107: MAP/Barometric pressure circuit low input.

Hope those are the right ones.

Or yeah 0420 not 0402 haha

Do you suggest doing anything prior to me taking it in for the test again such as driving it on the highway or anything like that?

Im talking after they are repaired. We already ordered the parts it should be fixed in the next day or 2. But as soon as its fixed I had planned on taking it down to get a second emissions. Some people say its a good idea to drive it a few miles even on the high way for a bit maybe even throw in some premium gasonline.