I think you've probably arrived in China on a Chinese Z visa.It is only good for entry to the country for the purpose of working here legally.It can get you a residence permit,which is compulsory for those with that kind of visa provided that your signed teaching contract is submitted to the local Public Security Bureau along with your foreign experts' certificate and passport.But if this is not done within 30 days of your entering this country,your Z visa will be downgraded to an L visa.You'll then be able to get a 10-day L visa,allowing you enough time to exit the country without being fined for overstaying - 500 Yuan/day up to a ceiling of 5000 Yuan. Please bear this in mind at all times.
As for your lack of contract,don't worry about it unless 23 days or so have elapsed from the date you enterd the country and it's still not forthcoming.In China it's commonplace to be in your situation as employers here are rather inefficient or like to do the necessary administrative work for the 2 or 3 foreign teachers at your school at one time rather than for each foreigner one by one.
And the fact that your head of department hasn't spoken to you about your demo lesson yet means nothing - remember 'no news is good news' in this context.I reckon your demo was at least average;if below average,you'd have surely heard about that at the time.
I've been in the same situation as you have at least 6 times.Each time I got my contract before the deadline.But if you're not good at teaching,you could have a problem.