katej0203

Get her UK passport.  
 
All of you fly to Canada / Mexico.
 
Then walk into the USA.   Tell the officer the kid is American. 

Yes. As far as I know, all GC extension receipt letters include an 18-month extension on your current card.

It is the best way to go.

I would personally renew the GC, that's what I did and now I am glad I did it because my N400 is very delayed (more than I expected).  At least I have a new GC that doesn't expire anytime soon and I can travel without any worries.

Yes you can do both. With pending I-90 they will have no problem issuing you an ADIT stamp (also known as I-551 stamp) into your foreign passport, the stamp serves as evidence of being a permanent resident for 1 year from issue and can be used for anything an actual physical green card can be used for (work, international travel, getting a driver's license, etc).
 
Also, USCIS wants you to turn in a valid green card or an I-90 receipt notice during a naturalization interview. So sadly you can't get around renewing that thing by just naturalizing instead.

Yes i90. They won't stamp you an i155 stamp without i90 receipt unless you filed for N400 6 months before the GC expiry. I90 filing doesn't extend your current GC.

Safest to do both.

Thank you, calling them was very successful. The lady on the phone has sent a request to the California Service Centre, and even given me a refference number for it. She said my "officer" should respond to me within 15 days. Certainly peace of mind hearing it from a person now, and also getting a refference number for the request. Phew!!