Confused about stay permit (to continue staying in China after the 2 month automatic extension)

Hi Marcellus,

Answering your questions to the best of my knowledge in-line to your questions:

-What do I need to do in order to apply for a 30 day stay visa extension once the two month extension expires?

• Visit an exit-entry bureau and apply for a stay permit in-person with documents listed below.

-do I need to be registered at a police station?

• yes, it is one of the required documents for the stay permit application

-will the stay visa 30 day extension cancel my 10 year L visa?

• [Confirmed with expert in the field] Yes, if you get a stay permit, you will be cancelling whatever visa you were on (in your case, your L visa). You cannot have more than 1 visa at a time. He said not to worry because, for Americans and Canadians, the 10-yr L-visa is easy to re-apply and get in their respective countries. 
• Stay-permit is currently the only option to staying longer in China. (Unless you get a job / study offer and successfully get all the paperwork required for a work permit / study permit).

-Is there anywhere I can exit and be able to come back without a mandatory quarantine if I am unable to get the 30 day extension?

• nope. 

-what documents do I need to provide?
Retrieved from https://gaj.sh.gov.cn/crj/enzwgk under stay permit tab…
“II. Documents required:
1. The applicant’s original, valid passport and visa;
2. An original and photocopy of the Registration Form of Temporary Residence for Visitors filled in with the applicant’s current valid passport and issued by a local hotel in Shanghai or the police station in the district of Shanghai in which the applicant resides;
3. A completed Visa and Residence Permit Application Form for Foreigners, and the applicant shall go to the local exit-entry administration of public security organs in person to take a free photograph, or the applicant who meet the relevant requirement of applying for visa without come in person shall submit one color identification photo (taken within 6 months, white background, 2-inch, bareheaded, without modification, photographic paper printed, without smears);
4. Supporting documents consistent with the reason for application:
(1) Visa-exempt foreigners shall provide application documents corresponding to the visa service notice relating to the reason for entry;
(2) Foreign seamen shall submit the seaman’s book (or other international travel documents), a verification letter from a shipping agency, and an air, boat, or train ticket with a confirmed date and seat, as well as family relation certificates for any accompanying family members;
(3) Those approved to renounced Chinese nationality shall submit their Certificate of Renunciation of Chinese Citizenship, foreign passport (or other international travel documents), and documents verifying the reasons for the stay;
(4) Foreigners whose reason for residence has been terminated shall submit proof of such termination;
(5) For foreign infants born in China, the applicant shall submit the original and photocopy of its birth certificate (including auxiliary pages) and valid passports (or other travel documents) of both parents;
(6) Relevant documents proving other reasons for the stay.
Notes: 1. Exit-entry administrations of public security organs may, as they deem necessary, require applicants to attend an interview or submit additional supporting documents (such as proof of financial support, proof of change of information such as name, or business operation certificate, etc.). Applications which fail to provide the requisite information within the required time will be rejected, unless caused by a force majeureevent.
2. All decisions of exit-entry administrations of public security organs rejecting applications for extension, replacement or re-issuance of ordinary visas, or rejecting foreigners’ applications for residence permits or permit extensions, shall be final.”
TL;DR:

• passport
• police registration (accommodation registration)
• application form (available at exit entry bureau)
• standard visa photo for all china visas
• a statement explaining why you need to stay longer in China (and, if applicable, no available flights to your home country… home country banning people coming from china etc.) 
• and any applicable supporting documents (printed screenshots of airline flight schedules, countries regulations during the coronavirus period, etc..)

Good luck, Marcellus!