To clarify the Z visa issue:
The Z visa is valid for a single one time entry to China in the period of 30 days after issue, for the purposes of employment. It does not allow you to remain in China for more than 30 days. It does not allow you to work or live in China.
It is normally issued outside of China and getting a Z visa may require a trip to your home country.
After entry to China, the Z visa allows a 30 day period to obtain a residence permit or leave China.
There no need to cancel a Z visa. It is used to enter China for employment but has a very short life.
Your Residence Permit is in your passport. I would not suggest you return it to anyone. It has an expiry date.
Your Foreign Experts Certificate, FEC, is, or was, a red covered book. Usually it is returned to your employer. In Beijing, I retained the same FEC as the employers arranged it. I think this, if all parties are willing, this is, or was, possible within the same jurisdiction.
In the olden days, the Residence Permit was a green book. They have not been issued since 2006 or before.
Warning: If you show up at a government office talking about extending, renewing, or modifying a Z Visa, you have just identified yourself as a visa idiot. You are then eligible for what ever the official's sense of humor has in mind at the moment.
Learn the vocabulary, definitions and jargon of visas and then ask the right questions. Search GoK and the internet. It is all there, free.