I got this job when I had my EAD but now I've had GC for couple months now... Do I have to show my employer my GC?
I would agree with those who say that you do not need to show them. The I-9 is an EMPLOYER's obligation, not an employee's obligation--and I think reading the actual law makes this clear. The employee's obligation is simply to remain authorized to work and to show I-9 documents when appropriately asked by the employer. But it's the employer who is supposed to initiate I-9 stuff--either at initial hiring or when the originally provided documents expire.
If there were a change in the nature of the work you are permitted to undertake, then you'd IMHO be required to show the new status. For example, if someone is on H-1B and gets an EAD or GC and wants to perform work for the employer that goes beyond the scope of the H-1B petition--then I think it would be required to update the I-9 even if the original expiration date on the H-1B hasn't expired. But if the nature of the job remains consistent with both the original work authorization and the current work authorization--I see no legal obligation to inform the employer of the change.
In terms of what I'd recommend, if you know for sure that you're going to be leaving the employer before the original EAD expires, I wouldn't recommend telling them. Just an unnecessary complication IMHO. But if you plan or hope to stay beyond the original EAD expiration date, why not let them know NOW? Eventually you'll have to make the update and why not reduce their stress level about your status by reassuring them of your new, more secure, US status? You aren't REQUIRED IMHO to update them sooner than necessary but nothing prevents you from doing so--and I think it's just good for your relationship with your employer to do so.
Finally, if your job requires you to travel internationally on business, it may be necessary or desirable to inform your travel department of the new status, esp if they need to arrange visas in third countries on your behalf. If the travel department communicates with the HR department--or if the travel dept and the HR dept are one and the same person in a small company--this may also trigger a need to update your I-9.
In summary--I don't think it's strictly necessary legally until the original I-9 date expires but it just seems wise unless you're leaving the company soon.