Depending on year and make of car. I believe the govenment requires that auto makers warranty the emmissions system for a period of 10 years, but this is easy for the manufacturer to get around b claiming non-warrantied faulty equipment. You may want to check via the internet your manufacturer and / or local DMV
No
Not if they test emissions.
Get it fixed.
No, the odometer is not part of an emissions test.
Replace the catalytic converter.
If it didn't have emissions controls from the factory you don't have to pass any emissions test, check with your state as some are different, especially California.
Nothing but do all the repairs necessary in order for it to pass. If you state requires this test, you have no other choice.
put a cat on it
No, it will not.
After one year, all new cars bought must pass the emissions test in counties that require the test. Once a vehicle is 25 years old, the emissions test is no longer required.
If it is leaking or rotted
Probably fix what's wrong.