Shelf life and preservation of Bánh chưng

How long can I safely store Bánh ch?ng in the fridge?

It depends on how good the bánh ch?ng was wrapped (before cooking) and if the bánh ch?ng is put into the fridge right after cooking and if the bánh ch?ng is already cut open. i assume that you have a halfway eaten bánh ch?ng. I would treat this case like regular cooked rice. A post on cooking.SE says that cooked dishes last 3-4 days in the fridge. StillTasty says 4-6 days. I just asked my mother. She said it will be fine for one week in the fridge if cut. You may have a look at the comments on the question "How long can I store cooked rice in the refrigerator?"

Can I preserve it somehow? I've heard that some people boil or freeze it.

Boiling would kill some pathogens. But since the toxins produced by the previously existing pathogens won't be destroyed, the bánh ch?ng would be still unsafe.

Freezing: I'm a student and hoard such stuff in my freezer :D Regarding food safety freezing is not problem. If you store the bánh for too long and/or don't wrap the pieces tightly enough the bánh will have freezer burn. I ate frozen bánh ch?ng which has been for more than a half year in the freezer. It tasted fine after steaming.
After defrosting you should either recook the bánh or steam it. I think you already knew this ;)

After unwrapping, bánh ch?ng can stay good for several days while a wrapped one can be kept for two weeks.

I think this statement sound quite plausible if the bánh ch?ng was wrapped very tightly and was wrapped in many layers statement and is stored in a fridge right after cooking. Bonus: The bánh was cooked in a pressure cooker. I guess, it's even sterile for some time o.o

some posts on online forums, where people said it can last even several months!

I doubt it, too. Did they really said "fridge", not "freezer"? o.o
_{(Oh, or their bánh ch?ng drowned in lard. Lard lasts for a very long time x))}

It depends on how it's made. Traditionally it is compressed under heavy weights after cooking and can last for quite a long time. These days they are often not pressed. Btw, that's why some banh chung you buy you'll find to be much denser than others.