They only need be registered as unemployed. The law [0] is in Russian but Google Translate does a good job [1], see 5.d
Free medical care in Russia is far from perfect, of course -- sometimes you have to wait for a month for an ultrasonic scan, sometimes they give you Russian insulin instead of Danish, but they will save you from a heart attack, install a stent and provide with necessary medications for at least a year.
[0] http://www.consultant.ru/document/cons_doc_LAW_107289/7e136d...
[1] https://translate.google.com/translate?sl=auto&tl=en&u=http:...
I hear about the EU a lot. It is nice to get some perspective from your part of the world.
Cheers
They go do the doctor as they would normally do and when they start working again they'll start paying their share.
That's how it is in Spain. Also if you don't work, you don't pay the drugs, at least some of them. But the subsidized part of the price depends on your income.
See reply above.
Greets from curious person in the USA.