I'll narrow my question down to Russia proper, but specifically in the late Soviet period, ~1970-1990. I know people often got housing when they married, so what happened to the housing if they divorced? And for their children-was there a concept of custody sharing, or were mothers just responsible? Did fathers party child support?
No, it wasn't really difficult, and people didn't get housing when they married (sometimes they did, but it wasn't a rule).
It was much more difficult before 1965 because from 1944 to 1965 there was a special law for divorcees. Divorce was relatively expensive (500-2000 roubles per divorce), and also to get the divorce you had to prove that your divorce is the best possible solution. And you had to prove it in court, there was no other way. And the thing is, there was no any list with 'good enough reasons' for divorce, so any judge could decide anything he wants in your case.
And if you're lucky, and you got your divorce, you had to put an advertisement in the local paper: just for your information, this man and this woman are divorced.
In 1965 all this changed, it was possible to get the divorce outside of the court (if you didn't have kids), but it still wasn't very easy, especially for communists. Not all soviet citizens were communists but for those who were divorce was a somewhat risky business. Divorce was considered immoral, and while you had this option, you could put your career at risk, and anyway your divorce had to be discussed with other communists or your colleagues, because they had to be sure that you weren't completely immoral person. Sometimes they could excommunicate you from the party, and it wasn't a good thing (it was much more dangerous and probable before 1965 but it was quite risky even after).
But for mere people it became much easier, and the process wasn't so painful as before. The only real constraint was that man couldn't divorce with pregnant wife and it was next to impossible to get a divorce if your kid is younger than one year.
The problem with housing was real, but it's not the 'people often got housing when they married' — this one is just not true. People got housing when they got housing, and they could wait years and sometimes 10-15-20 years for their apartment. But it was a problem, because there wasn't a real market for apartments, it was hard to buy or to get a new apartment, there weren't credit lines, et cetera. As a result sometimes divorcees continued to live together for years just because they didn't have other options or didn't have money to exercise those options (someone could rent a room if he/she had money or they can't exchange their 2-bedrooms for two one-bedrooms and so on).
Regarding kids — I believe that this haven't really changed. Even now there isn't really working concept for custody sharing. As a rule mother get the kids, and father has to pay for them (25 per cent of his salary). All the decisions are usually made by mothers but fathers have the right to communicate with their kids several times a month. In most cases it works okay, but there are no strict rules and in edge cases it's up to the judge.