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> 1. My mom's multi-entry visa was expired back to the beginning of this year. As she is in the queue for her GC, that B1 visa is no longer matters, right?

Correct. She should not use a B visa to enter the US, as that will abandon her I-485.

> 2. How should my mom use her AP? Does she need to show it to the custom officer when she leaves US or just when she comes back to US? It is just a letter, will it be aquired or taken away by the custom officer when she comes back to US? As its expiration date is in 5 years, how many times could it be used? Is it still valid even after my mom gets her GC? 

The US has no exit immigration checks. Nobody encounters any officers when leaving the US. The AP is used when they come back to the US.

AP is usually multiple use (meaning they can use it as many times as they want until the expiration date). It should say on the document.

I read somewhere that for paper AP, they get two sheets of paper, and on the first entry the officer takes one of the copies and returns the other copy, and for all subsequent entries they just show the remaining copy and the officer doesn't take it. I am not sure if this is still how it works.

Once they become a US permanent resident (i.e. green card holder), they have unlimited right to enter the US, and the AP becomes useless.

> 3. Any other docs/files my mom should carry with her for her urgent trip?

It would be good to carry the receipt of the pending I-485, so the officer can verify that the I-485 is still pending when they enter on AP.

> then on, the wait time always says 4 months. Just in case if her GC gets approved when she is at China, even it is quite slim, what should we do? Does she need to wait I mail her GC for her to use when come back to US or she can use AP to come back?

If that happens, they can still use the AP to board the flight back to the US. When they reach the US port of entry, they can tell the officer that their I-485 is approved, so that the officer can look it up and admit them as a permanent resident.