2001 Buick Century engine rough idle

There is a tsb for this but hard telling if your computer has been updated or not. But check for a stuck open or partially clogged egr and fuel pressure it may be low 41-47 psi autoparts rent gauge

If the egr valve was stuck open would the engine stall when get of the gas and will the engine code P1404 causing the engine code PO171 and what is the meaning of the PO171 and P1404?

Yes because an egr open at idle cause a rough idle. 171 is lean engine and 1404 is an egr problem.

The engine do not stall when take foot of gas pedal so which one do I need to replace the p1404 or the po171? To me it seem like the engine is more like a rich engine.

You need to fix both not just one, check your fuel pressure as well.

Are you saying I need to replace the Oxygen Sensor and the EGR to fix it?

You need to find the cause of the codes which can be other than what you think they are the p1401 is going to be a bad one to fix because it may need a pcm upgrade if it never has been done. You can change theO2 sensor but that may not fix the code you are getting. But you still need to check fuel pressure with a gauge as that can also cause the 171 code. Replacing parts doesn't always fix the code problem