To what extent were charities allowed to organize and operate in the USSR?
Might a group of like-minded citizens be able to come together and raise funds for, for example, the plight of stray cats or research into treatment for a medical condition, or was this sort of activity prohibited by the state?
The short answer is no, like-minded citizens couldn't come together and raise funds for their own objectives. There were no financial and legal tools for that (you couldn't open a bank account to collect money), you couldn't register non-commercial entity (and you couldn't register a company as well), and, more than that, charity was seemed as something bad and non-appropriate for the socialist or communist country, because there you don't have to need charity at all.
As far as I know, charities were not forbidden per se, but if you tried to collect money, you would end up in jail because you had to broke a lot of laws to do that.
"Charity is needed for capitalist countries, because with this capitalists are able to cheat simple workers so these workers wouldn't want to fight capitalism" — this is a very rough and shortened translation of the beginning of 'Charity' article from the Big Soviet Encyclopedia (1950). Obviously, it wasn't so rough in 1970-1980, but you still couldn't raise a charity.
At the same time, you sometimes could (and should) spend your own money on charity. If this charity was backed by the state. Mostly, these charity funds tried to help some foreign country workers or something like that. One of the most popular was Soviet Peace Foundation (it seems that their objective was to make the world more peaceful than it was). I don't think they had published any financial stats but one of its directors, famous Russian chess-player Anatoly Karpov, said that its assets were, like, seven billions dollars in 1989. Some of these billions were collected from school kids who had to give their pocket money away.
Sometimes the money were collected for something slightly more tangible: to help workers in (choose your own) country.
You couldn't legally raise money for your poor neighbours or hospitals et cetera, because there were no poor people in the Soviet Union, and hospitals were completely okay. You couldn't raise money for your own objectives. But you could spend your money on state-owned charities to help your foreign comrades and to make the world the better place. And, sometimes, you were demanded to do this (even if you didn't believe it or just didn't wanted to help anyone).
I'm not sure if this is true for early years of the Soviet Union (1920-1930). It seems, there were some charities at that time (there is a scene in 'Dog's heart' by Boulgakov where some guy is saying that professor Preobrazhensky should give his money to hungry german kids), but I don't know anything about that.