Great question. I don't claim to be an expert on the subject, but I'll shoot from the hip.
I'd be hesitant to call it communism, or even socialism. There is a vast difference between those political ideologies and the religious principle we call the Law of Consecration (united order). It is basically a commitment members make acknowledging that everything they have comes from God, and that if needed, the member would give everything they have to the church for the building up of the kingdom. It has been used a few times in church history, but typically in times of severe need.
Currently, members are asked to pay a tithe (10%) of their increase. In addition to this, they fast (no food or water) one day each month and donate the funds that would have been spent on that food and donate that as a "fast offering" to the church. These funds are used in different ways. Tithing pays for things like buildings, maintenance, fleet vehicles, printed materials, activities, etcetera. The Fast Offerings are used 100% for helping the needy. There are no paid clergy in the church at any level, so there is zero overhead.
I think that Randian Objectivism might be a bit of a stretch. While there are clearly some Randites in the flock, I think the vast majority of mormons just believe in hard work. They work hard, earn a living and don't like a government encroaching on their income with higher taxes. They give their time and money to create a safety net with little to no waste. It's hard to see the government with huge waste. You'll have a hard time finding a group of people more patriotic at their core. All this despite a touchy relationship with the government at times (in the state of Missouri you could have killed me for being a mormon up until 1976).
I don't know if I really answered your questions well, not being an expert in church history, nor in Rand. I think perhaps the most important thing to realize is that members of this church are... just that: members of a church. It's a religion, not a political party. You'll find democrats, republicans, communists, socialists, feminists, and probably almost any other *ist you can think of. Even within my family you'll run the gamut on social/political issues. The current 12 apostles don't see eye to eye on every issue. This church is made up of all kinds of people.