Short answers, pulled from the story.
Square was founded by Masafumi Miyamoto as a software subsidiary of Den-Yu-Sha, an electric power conglomerate led by his father Kuniichi Miyamoto. The company began operations in a former hairdresser's salon in Hiyoshi, Yokohama.
Square's first completed game was The Death Trap, which was released in 1984 for the NEC PC-8801. This text adventure set in a war-torn African nation was the first game published under the Square brand.
Square moved its headquarters to the Shimomeguro district in Meguro in 1995. This relocation occurred after the company had previously moved to Ginza in April 1986 and later to smaller offices in Okachimachi, Taitō.
Square Enix was formed on the 1st of April 2003, with Enix as the surviving corporate entity. Square dissolved its departments and subsidiaries into the new company, and around 80% of Square's staff transitioned into Square Enix.
Final Fantasy: The Spirits Within cost Square and co-producer Columbia Pictures US$137 million and was released in 2001. The film grossed only $85 million, leading to a financial loss for Square and the closure of Square Pictures.
Square was listed on the Tokyo Stock Exchange's first section in August 2000. This section includes the largest companies on the exchange and marked a significant milestone in the company's history.