I was pondering about the 1982 war over the Falkland islands, formerly the Malvina islands belonging to Argentina, and the Sandwich islands as well. It seems a little hypocritical of their government to allow other formerly occupied nations to have their sovereignty, no matter how badly the aftermath destabilized some of those nations, only to turn around and steal land from others. I understand some areas have strategic value, but couldn't they have come to an agreement like building a naval base there instead of just starting a war and stealing land from another prominent nation? How did they get away with this without any repercussions or sanctions from the UN?
Could you possibly clarify what you're after here? The UK has claimed sovereignty over the Falklands since 1765, and British subjects lived on the islands continuously from 1833. Its claims are certainly contested, but any "stealing from another prominent nation" that may have taken place long antedates the decolonisation process that began in 1946-47 and the existence of the UN.
Similarly, whatever your take on the events of the 1980s, the British were both the occupiers of the islands prior to 2 April 1982 and not the party directly responsible for starting the conflict there.
This 1983 paper from International Affairs summarises the sovereignty claims made by Britain, France, Spain and Argentina, and assesses their standing in international law, both as it applied in the C18-19th and under the modern, UN-endorsed, principle of self-determination. It also contains a summary of the history of the islands.