Most universities and schools have outdoor, permanently installed, ping pong tables.
If you turn up with two bats and balls, you will quickly get a game from random students passing by. There will probably be others playing. People on campus will just assume you are another student or teacher.
And its FREE.

You won't get a teacher, but plenty of practice is most important for most games. You can check the rules online if you are unsure.
You can also advertise for activity partners on here.

If your aim is just to do something, there used to be a lot of ads on here for Ultimate Frisbee, and bike rides.

This is a hard and sad case, but the rules sometimes fall harder on some than others. Governments often have blanket sets of rules which are easy to understand (by all) and easy to apply (by government officers). The downside is that there are 'hard cases'. To change the rules for a few hard cases, causes knock on problems. Also from an enforcement point of view, you start making exceptions and there will be a queue of people trying to be an exception. Managing these types of cases adds significantly to the workload of already overstretched government staff and is in itself a drain on taxes.
Its tough, by I do understand why cases like this occur.

Keep an eye out for rooms here. Some are advertised in advance. I have seen prices for a room vary, between 700 (couple or years back) to 1500/m.

It also depends on where your training is. Some parts of the city are more expensive.
Rooms don't come up too often, but more so at the beginning of the academic year. Most rooms I see advertised are looking for female room mates.
Some of the private schools do have campus accommodation, or can help you find somewhere to live. All unis have campus accommodation.

Eating out, can cost from about 10rmb for a simple bowl of noodles. Then it jumps to 10-20 per dish, plus 2-5 for a bowl of rice.
You may also do pizza and beer when you need it. Pizza from about 30, beer from about 10 (eat out prices) for local beer.

Toilettes seem to be a similar price to UK.

There are websites like teach-in-china.net and Daves ESL cafe that list jobs China wide. Now is about the time that most schools/unis recruit for next year.
The likes of the big chain schools have a mixed reputation. Mostly because they are franchises. Some of these owners are out and out skunks. A few are OK.

The private language school pay is higher, but they provide no accommodation, flights, and some do not want to provide a work visa. Most want to employ people already here on some other visa. The hours are usually evenings and weekends. But your weekdays are often free until about 3pm, unless they stick you with office hours. You will usually do 16-20 hours teaching per week, and often office hours, which takes you up to 40 hours in the centre. Basically you will work from lunch time until 8pm or 9pm.
In the public sector, the pay is lower, but they provide private accommodation, flights, basic medical cover, and a work visa. Public sector jobs are M-F only. 16 hours teaching maximum (but usually the full 16). However when you are not teaching your time is your own. Teaching hours usually fall between 8am and 5pm. University hours are usually 8-12.