Yes, it is very shady and illegal. It puts all the burden on you--you will be living and working illegally in the country, and there is no paper trail to them. That way they don't have to spend any money on you, they can test you out and if you get picked up by the police (which, by the way, in June a big crackdown on foreigners working illegally is in the works) you get jailed, fined, thrown out and your passport blacklisted, and they get away scot free. 

DO NOT do this. There is no legal reason for them to deny you a Z visa. It if is truly "reasons beyond their control" this means one of three things: First, you don't qualify for a Z visa (you don't have the proper degrees or you haven't worked at least 2 full years in your home country after getting your degree) and they need to grease some wheels and pass out some bribes to try to get it done under the table. Second, they don't have a license to employ foreigners, and either are in the process of getting one or are trying to find someone with whom they have a relationship to pretend to be your employer so they can get the visa. Third, they have no intention of ever getting you a visa, but want you to work for them for a few months at a lower cost. 

Be aware that if they do eventually get you a visa, the process will take at least 2 months, during which time you will have to exit the country. You'll have to at least go to Hong Kong, and very likely back to your home country. They will certainly expect you to pay for this trip on your own. Then you'll get back and it will be weeks more before you can get your residence permit. So the actual time that you are working illegally could be extended as far as six months. 

Don't do this. If you think they're just negotiating with you, tell them you'll only work for them when you are here completely legally on a Z visa, and that what they are asking you to do is  criminal. If they shove you aside at that point, report their demands to the labor bureau. Won't do any good, but one can always try. 

What kind of job would you do? Anything being 'foreigner paid in cash' is shady, so amount of cash should be much higher than standards.

Let say for example, you accepted 'English teaching' in the city at one School. Suddenly, your employer wants you to teach in another city, with more hours and same pay. 

If you feel violated by your employer, you can seek help or explanation at Foreign Expert Bureau, but you must be legally employed to receive Government Agencies help.

Same as you cannot seek protection of Police, if you're getting robbed while selling fake cigarettes or alcohol on the streets.

If you would teach for two months with cash payments, your job would be similar to part-time jobs. If I work part-time at Kindergarten or I have private classes in my home, I must get my pay weekly. Hourly pay for my part-time work is always higher than pay at full-time job.

In my experience it is pretty common and there are plenty of cases where it does work out fine.  But also plenty of cases where you get screwed over.

Like others mentioned, companies do this knowing the burden is now on you and just the fact that they can get you to agree to something illegal early lets them know that you are probably very easy to manipulate later on.

My advice to anyone working in China is that if you feel something is shady... DONT DO IT.  It's better to be safe than sorry.  I see too many naive foreigners who know something is shady or illegaly but they still "trust" people are too much and "hope" things will work out.  You cant "trust" or "hope" much when doing business in China, you have have your requirements and not budge at all, period.

Good job walking away, if the company is/was serious about hiring you and they intend on treating you well then they will do something to get you to come back and work for them, if they don't its clear they are just looking for the first naive person to fall for their tricks.