The original May Day was a strike held in Chicago in 1886. The strike started on May 1st. At the time there was no specific significance to the date, yet. The police responded and the companies called out strike breakers. On the 3rd somewhere between two and six strikers were killed when the police fired into them. On the 4th a massive protest against the police violence was held. During the protest someone threw a homemade bomb at the police and the protest went downhill from there. Contemporary reports are all highly charged and vary based on the political views of the reporter. Either the crowd opened fire on the police or the police opened fire on the crowd. Either the crowd had guns and more bombs or they didn't. Even today this is sometimes called the Haymarket incident, riot, or massacrer. The end result was seven dead police officers, four dead protestors, at least 60 wounded officers, and several times as many wounded civilians. A number of local members of anarchist groups were tried, convicted, and some hung for initiating the incident.

The date became an important memorial date for members of the international socialist movement. And when the Second International decided to hold universal strikes across the world (mainly in Europe) in 1890, May 1st was chosen as the day specifically to honor "the martyrs of Haymarket." In Europe the reporting about Haymarket had generally been more positive towards the workers. So when it came time to establish an official "Labor Day" most of Europe May 1st was an easy choice. It had become an unofficial date well before it was officially recognized.

In the U.S. though, the date was fraught with a lot more bad blood, literally. So rather than allow that date to be memorialized, the U.S. adopted a date that commemorated a public workers strike in NYC. There had already been a number of state holidays, which weere held throughout the year. So when Congress decided to create a Labor Day, they had a lot of choices and early September was already one of the most common. Besides May 1st wasn't seen as having a lot of positive associations.

To answer you other question the U.S. Was establishing Labor Days as early as 1887. And the federal September date was chosen in 1894. The first May Day was held in 1890. The big international May Day riots happened in 1894. And May 1st wasn't officially adopted by socialists as the International Worker's day until 1904. So the two holidays developed in parallel. Grover Cleveland didn't pick September to avoid association with international socialism. But he did pick it to avoid the event the socialists were commemorating.

Most of America does celebrate Workers' Day or Labour Day in May: North America has September - a product of its own union history, as explained above.