work permit and residents permit transfer questions

I am thinking about leaving my current job and transferring to a new company. I will sign the new contract before handing in my notice at my current company, but I'd like to know how long I have to submit the documents to apply for a transfer. I understand that if I don't transfer quickly enough, I will have to apply for a whole new visa rather than simply transfer.

My resident's permit will also still be valid until Jan next year, will this need to be cancelled and a new one applied for with my work permit or can I keep it and just reapply when it expires? If I can keep it, will I be able to travel in and out of China during my work permit transfer or is it not permitted to travel with just the residents permit and no work permit?

For the visa transfer, will my old health check (a year old) and authenticated degree certificate (authenticated a year ago) still be valid or should I look at getting them redone ahead of the transfer application?

Many thanks.