King Francis' death on the show mirrors how King Francis II died in real life, including a seemingly random collapse, followed by quick deterioration, all brought on by an inner ear infection. Like on the show, he had been King for under two years at the time of his death.
While the show doesn't tell in real life several days after Darnley's death Lord Bothwell "abducted" Mary and the two ended up getting married. They were married long enough that she got pregnant by him with twins, but she suffered a miscarriage.
When Elizabeth meets with Francis Drake for the first time he calls Sir John Hawkins his friend. They were friends, but also cousins as well.
When Catherine goes to get her daughter Margot she is credited as Margurete de Navarre. She was not that at that point in time though. The only way she was of Navarre was because she married King Antoines son. The marriage was was tumultuous leading to a divorce and him marrying Catherine's distant cousin Marie di Medici.