No private hospitals but some specific areas of medicine are provided for fee/profit when not covered by public health care (see a more detailed response below). As mentioned, mental-health services are covered but the availability is woefully inadequate practically everywhere through public channels, due to a shortage of resources.
Almost no private care facilities here; with the exceptions I've mentioned and maybe a few more. It is meant to discourage a separate system that would perhaps be more lucrative for health-care professionals. Thus they might be drawn away from the public system, in which the delays/waits are in large part, due to the lack of human resources. There's a few different ways this can be argued; pro-and-con, but there it is.
It's a percentage of insurable (not gross) income, so it's a bit of a complicated formula that varies. Employers also match the amount.
Yes, but like everything else, there are limits to the resources and with mental health/addictions services in particular, the need HUGELY outstrips the meaningful care available in a timely manner. These services are also some of the few that can be legally provided on a fee-for-service basis. One can access a psychiatrist, counseling or in-patient rehab services tomorrow if they have the cash or private insurance.