Yes they do. In fact, many of them see their business increase during that time and a lot of foreign teachers who work at university supplement their income during this time by working in these teaching schools.

Depends if it's private training center or a state school. If it's a state school, then no. There are also many English summer camps where the poor kids have their holidays stolen away from them over intensive training camps

I teach in a Language Training Centre and yes, the 'school holidays' are our busy period as those who didn't get the gaokao marks they wanted cram English for IELTS testing to go to 'foreign' universities, (of course this only applies for those with wealthy parents).