Re: Immigration permit for countries such as Korea
You would probably have to ask about each specific country, as I am sure each country has its specific laws with regard to immigration.
If it was me, I would look up on Google each specific country and see what their specific requirements are. Personally I have no idea what a "care and control" case is. My bet is you would have to contact the relevant authority in each country you are considering to ascertain whether you are eligible or not.
I will say that over the years I have investigated immigration to a number of different countries. I can not speak to the aspect of having a criminal record, but I can say all had some type of financial and/or other requirement such as investing in a business there.
If it was me and I had the money I would hire an immigration lawyer as the paths to entry vary even in an individual country. Most countries have immigration lawyers who can help - Of course, you're talking paying them for their services.