It is a requirement to have a physical examination throughout China to be able to complete the process for an invitation letter for a 'z' visa. In some instances you can miss out the chest x ray and ultrasound which should reduce the cost slightly.

You can download the form here, http://www.china-embassy.org/eng/ywzn/lsyw/vpna/rap/W020110807201675371788.pdf

You might be able to find a Doctor who will stamp and certify these credentials at a cheaper price but you will need to send something to a prospective employer first. The physical will have to be redone at the designated foreigners hospital in the provincial capital where you intend to work. This is because it must be certified by Chinese doctors. There is no workaround on this.

Your consulate may not require the document but the school would require it to be even issued with the invitation letter in the first place.

To be legally employed in China you need a 'z' visa and associated residence permit which requires the medical exam.

As for the six month rule, you are correct. Generally schools are unwilling to offer a contract for six months as the initial cost to them in recruitment and the costs involved in procuring you the 'z' visa and resident permit can well exceed 10,000rmb. It is not cost effective for them to do so. I say this as someone who is involved in recruitment for my school (we do not use agencies) and see first hand the ever evolving rules and regulations that we must follow.