You need a 'Z' Visa to enter China for the purposes of work. It is then converted into a Residence Permit that replaces the 'Z' Visa allowing you to work and reside in China.
Once you have a Residence Permit it can be renewed annually. The 'Z' visa is no longer necessary.
If your Residence Permit has expired, you need to start the proces over again, meaning obtaining a 'Z' visa outside of China to be converted into a Residence Permit once here.
The process of converting 'L' and 'F' visas into 'Z' visas is not supposed to be done, however in the past it was allowed. Now it seems the letter of the law is being held to and conversions of this type are not being issued.
What is happening now is not a change in the rules but a more strict adherence to them.
I have had no 'Z' visa for 4 years but have been legal due to my Residence Permit.
The Foreign Expert Certificate is to show that your job (which is essential for your company) cannot be done by a Chinese National and so your company has had to hire outside of the country. This certificate is tied to the company. A company without legal standing to hire foreigners cannot obtain it. If your Foreign Experts Certificate has the name of another company on it, then you have been 'loaned' out. The legality of this is murky at best.
The Certificate of Health Examination is needed to obtain both the Residence Permit and the Foreign Experts Certificate. It is to make sure you are not introducing any foreign nasties that could disrupt the harmonious healthiness of the people of the Middle Kingdom.
In the past there have been many work arounds but now this is what you should be looking for.