Want to find out if I must get a Z visa, work permit, and residence visa? See detail below

You are not required to show earned income in China to live and work. You need to get a Z visa, work permit and residence visa to work in the China branch office of the US company.

If you are the chief representative of the branch office of the US company, the branch office should take the the registration certification issued by the administrative authorities of industry and commerce to apply for Invitation Letter of a Duly Authorized Unit, then send it to you.

You take the Invitation Letter of a Duly Authorized Unit and the copy of registration certification to apply work visa at China embassy/consulate in USA, then you may apply directly for the Employment Permit by presenting your work Visas(Z) and relevant papers after your entry into China. At last apply for residence permit (it can be called residence visa)