Short answers, pulled from the story.
The Ptolemaic Kingdom was founded in 305 BC by Ptolemy I Soter, a Macedonian Greek general who had been one of Alexander the Great's closest companions. Ptolemy had governed Egypt as satrap from 323 BC before taking the titles of basileus and pharaoh in 305 BC.
The Ptolemaic Kingdom ended in 30 BC with the suicide of Cleopatra VII on the 12th of August and the subsequent capture and execution of her son Caesarion. Egypt's fall resulted from entanglement in the Roman civil war between Octavian and Mark Antony, culminating in the naval defeat at the Battle of Actium.
Cleopatra VII was the last ruler of the Ptolemaic dynasty, ascending the throne on the 22nd of March 51 BC. She was the effective ruler of Egypt despite having nominal male co-regents, and she allied first with Julius Caesar and then with Mark Antony in attempts to restore Ptolemaic power against Rome.
The Library of Alexandria was a major research center founded by Ptolemy I Soter, possibly with advice from Demetrius of Phalerum, located in the royal sector of Alexandria. It held several hundred thousand volumes and supported scholars including the mathematician Euclid, the polymath Eratosthenes, and the poets Callimachus, Apollonius of Rhodes, and Theocritus.
The Ptolemies adopted the Egyptian title of pharaoh alongside their Greek title of basileus and had themselves depicted on monuments in Egyptian style and dress. They built and restored temples, patronized the priesthood, participated in Egyptian religious life, and created the god Serapis to unite Greek and Egyptian religious traditions.
The Rosetta Stone was a stele commemorating the defeat of the Great Revolt of Egypt, a native uprising that lasted from 205 to 186 BC and nearly drove the Ptolemies from power. Like other Ptolemaic decrees, it was inscribed in hieroglyphs, Demotic, and Koine Greek.