Short answers, pulled from the story.
Cyrus the Great defeated Median king Astyages and captured Ecbatana in 550 BC. This event marked the birth of an empire that stretched from the Balkans to the Indus Valley.
Persian forces defeated Pharaoh Psamtik III at Pelusium in 525 BC, leading to his flight to Memphis and subsequent suicide. Egypt became part of the Achaemenid Empire while garrisons like Elephantine remained stationed there throughout his reign.
There were between 20 and 30 satrapies across the vast territory at varying times. Each satrapy had a governor called a satrap who administered the region with oversight from royal inspectors and state secretaries.
The naval Battle of Artemisium occurred in spring 480 BC followed by the decisive Greek victory at Salamis later that same year. The final defeat for Persia came at Plataea in 479 BC after troops retook Athens but were destroyed.
Alexander marched on Susa and Persepolis which surrendered before burning early in 330 BC. Darius III sought refuge at Ecbatana but was captured by Bessus, his Bactrian satrap, who murdered him and declared himself Artaxerxes V.