Yes, it will loose power and use an excessive amount of fuel.
The catalytic converter is most probably broken
No.
Yes it does.
Not likely but possible.
have the car checked out, it could be a bad o2 sensor or a plugged converter.
Yes. Oil is hard on Catalytic converter.
no it will run rough and stall out
Not likley. You will normally see a loss of power and the converter will get red hot.
It is one of many possibilities.
catalytic converter
Yes, a bad cat can cause issues.
Yes, if it is clogged very badly.