Short answers, pulled from the story.
Serbia played their first match as a fully independent nation after Montenegro declared independence on the 3rd of June 2006. The team's first official fixture under the Serbia name was a 3-1 win over the Czech Republic. FIFA and UEFA recognise Serbia as the official successor to the FR Yugoslavia and Serbia and Montenegro national teams.
The Federal Republic of Yugoslavia was banned from all international sporting competition due to United Nations sanctions stemming from the conflict in Yugoslavia. Despite qualifying for Euro 1992, the team was barred from participating, and the ban also applied to the 1994 World Cup and Euro 1996 qualifying process. The team did not play its first match as a new country until the 23rd of December 1994.
Serbia, competing as FR Yugoslavia, reached the round of 16 at the 1998 World Cup in France. They were eliminated by the Netherlands when Edgar Davids scored in the final seconds of the match to win 2-1. At Euro 2000, the team went one round further, reaching the quarter-finals before losing 6-1 to the Netherlands.
Dejan Stanković holds the record with 103 appearances for the national team. He surpassed the previous record of 102 caps held by Savo Milošević. Stanković's final international appearance was a 2-0 friendly win over Japan.
Stanković became the only player to appear in a World Cup having represented three different national team names: SFR Yugoslavia, FR Yugoslavia, and Serbia and Montenegro, before competing with Serbia. This was a result of geopolitical changes rather than any change in his personal nationality.
Serbia qualified for UEFA Euro 2024, their first European Championship as an independent nation. They finished second in qualifying Group G with 14 points. It was also the country's first appearance at a major European tournament in 24 years, their previous participation having been at Euro 2000 when they competed as FR Yugoslavia.