I was amazed by the amount of open gay activity I saw in China as far back as 2004. Bath house was tacked onto a major gym in central Shanghai, Gay Mr China competition in Bejing, men doing their preening in public changing rooms.
People may not be able to come out to family, friends and employers; but being gay is far from being underground in China.
I found that for me, once I obtained the certificate, it was easier and cheaper to get an agent to the admin.
The process in the UK (I assume US is similar) is as follows.
1/ Get certificate
2/ Send certificate to be validated (proven to be real) by the Foreign and Commonwealth Office (UK equivalent of State Dept). Only notorization by FCO is recognised by China embassy in UK.
3/ Send notorized/validated document to Chinese Embassy to be certified (certificate is attached to rear of doc by Chinese authorities) and stamped (chop).
4/ Send this document with other required documents with Visa application.
I used an agent as it saved me a trip to London and 5 days of running around. Like I said, I assume (but could be wrong) that a similar process is universal.
In China you only pay tax on the income that is declared, by either you or your employer. It is not unheard of for an employer to 'deduct' tax and national insurance from pay and then keep it themselves.
In China, if you are over retirement age (60), you will not get a work visa. You will not get recruited by the public sector schools and universities, they are not permitted to employ over 60 foreigners any more.
Many teachers returned home when the regulation was brought in a few years ago. However, some teachers of 60+ years end up doing something in the private sector; perhaps on the wrong visa. Others freelance as tutors (cash in hand), also technically illegal.