Short answers, pulled from the story.
A flamen was a specific type of priest in the ancient Roman state religion, dedicated to the cult of a particular deity. There were fifteen flamines in total: three senior priests called flamines maiores serving Jupiter, Mars, and Quirinus, and twelve lesser priests called flamines minores serving various minor deities.
The flamen Dialis and his wife were forbidden from touching, seeing, or referring to yeast, raw meat, goats, dogs, ivy, or beans. He could not ride horses or leave Rome for more than a single day, and was barred from holding any political office. Their marital bed's legs had to be covered in clay and could not be slept in by anyone else.
According to the historian Livy, the three major flamines were created by Numa Pompilius, the legendary second King of Rome. Numa established the offices to fulfill the intensive religious duties that had previously been performed by the King himself, granting each flamen a fine robe of office and a curule seat.
Mark Antony became the first flamen Divi Iulii in 40 BC. The Senate had approved the creation of this fourth major flamen two years earlier, in 42 BC, following the assassination of Julius Caesar, officially recognizing Caesar as a state divinity.
The flamen Dialis Lucius Cornelius Merula died by suicide in 87 BC, and the extreme taboos attached to the office made it nearly impossible to fill. The vacancy lasted for over 70 years until the Imperial reign of Augustus. A teenage Julius Caesar was once nominated to fill the position but did not take it.
After the Lex Domitia de sacerdotis was passed in 104 BC, nominations for the flaminate were made by the Tribal Assembly and reviewed by the pontifex maximus. The newly chosen flamen then had to undergo a ceremony called the captio on the Capitoline Hill, during which an augur sought the relevant deity's approval through the reading of divine signs.