Camia07

Well for the ones that find themselves in the same situation, here is the end of the story. 
After multiple phone calls where no one was able to help. I went in person to the American Embassy in Cairo, Egypt.
Once at the gate, they refused to let me in as I’m not a American citizen but let my husband go in for me (which was kinda weird because he isn’t the petitioner. But whatever. ) he talked to many people there, they looked at my passport / visa / emails etc, all this to let him know 2hrs later they can’t help. 
Useless embassy, waste of time.

I ended up canceling my return flight to the US and went back to my home country and stayed there awhile until I received the new visa.

Cool, don't forget that with the I-131A, I am fairly sure you need to pay before taking it to file (as written in the 'where to file' box on the link).
 
 
You had your I-551 because you filed an I-751, right? And the extension letter ran out, and you got an I-551 stamp, correct?
Then yes, you are an LPR. A conditional LPR, technically, but thats semantics as a conditional LPR is an LPR. 
 
 
The stamp is evidence of your status, not the status itself. Only an immigration judge can terminate your status (or yourself by filing an I-407, which I'm fairly sure you haven't done). 
 
Well, since there is no 'I'm an idiot who left the US without having proof of status and without consulting VJ first' option, you would select the first one, which is'
"Returning from temporary travel outside the United States of less than 1 year, and your permanent resident card (also known as a green card) has been lost or destroyed".
It technically hasn't but it would result in the same situation you are in - being stuck overseas with no current proof of status. Thats the closest option to your current predicament, the same as if you lost/someone stole your passport with your I-551 stamp in. 
 
The second option is applicable to those LPRs with re-entry permits (confusingly called an I-131, which is very similar to the permit you need which is an I-131A, but is a completely different thing),  who have been (legally) out of the US for more than one year and less than two years. This is not applicable to you. 
 
The third is for those who aren't LPRs and are travelling while on EAD, advance parole and other such similar status. This is not applicable to you.
 
 
Yes. Again, The stamp is evidence of your status, and not the status itself. You are a conditional permanent resident. You are a permanent resident up to and beyond that being adjudicated. Only an immigration judge can terminate your status (or yourself by filing an I-407, which I'm fairly sure you haven't done). 
 
 
 
In some ways it is designed to be confusing and opaque. How do you think these clowns make money? 
Yes, you do now know more than the agent, but stay humble. Be assertive, and firm is necessary, but humble, and remain calm.
You are a LPR, you have rights enshrined in law, and yes, you now do know more than the average person, thanks to this expensive lesson. 
 
 
Remember, there is still always the option of the cheapest flight to Canada or Mexico and entering via a land border.
Book the cheapest return you can find, to Toronto, Montreal, or Vancouver, find your way to the border, and then when in the US throw away (forget about) the return flight - it would be cheaper than booking a one way. 
 
 
 
 
*yes I know the OP is not an idiot and it is not a personal attack.  I'm posting it to emphasis the very real seriousness of their predicament. Hopefully it will assist someone else in future who searches for it, and prevent them ending up in the same situation. 
 
 
 

Yes. Only the evidence of your status has expired.  You are a legal resident until an immigration judge says otherwise.

I would love to know to whom you spoke.  Call back and give the person this link from the US Embassy in Cairo:
 
Boarding Foil - U.S. Embassy in Egypt (usembassy.gov)

Permanent residents cannot get visas or ESTA to the US. Visas are for foreign travellers without permanent residence.

HUCA - hang up and call again. Not all agents are created equal. 
 
The info on an I-131A (commonly known as a boarding file) is HERE . You pay and fill in the form before taking it in person to the embassy - the instructions are there in that link. 
 
 
This alone demostrates the person you spoke to knows absolutely nothing.
Permanent residents are not eligible for ESTA, and if you tried, would more than likely be denied, and if not, it would almost certainly get picked up by the API.
This is not an option for you.