There's been no documented account of skull and swords occurring. The Death's Head and cross bones, yes. Some other designs, yes? All of those other ones you've seen printed in books - surprising, more flag images printed in books as "real" are not. I'll let /u/ETFox fill in more, he's studied them before.

EDIT: As for who flew it first, so far I've heard that was attributed to Emanuel Wynn around they year 1700.

As for origins, well, the origins of the designs that aren't documented that are so popular now may go back to the Philip Gosse papers of the early 20th century. As for the name "Jolly Roger," while people have various conjectures about it being french or a reference for the devil - the only hard evidence from the era, and the earliest available, is an account from the Weekly Packet on December 12, 1719. It says that the name was inspired by pirates referencing Woodes Rogers, the new governor of the Bahamas that fought against them. The pirates said, "then they would go into Providence, to their old Friend Johnny Rogers, and lay down their standard, which they hoisted topmast head, with a Gun and Sword, which they call’d Johnny Rogers." Eventually, this got turned into Jolly. I forget why, but Johnny Rogers was a colloquial nickname for for Woodes Rogers.

I would suggest /u/davidAOP, answer...

Wait, I mean, u/ETFox.

Wait.

Okay, fine....

Specifically the idea of the skull and crossbones is an old one that predates even the opening of the New World. It can be found on tomb markings as early as the late Middle Ages, and generally means simply "death."

The first literary mentions of the Jolly Roger appear in In Captain Charles Johnson's A General History, where he mentions a relatively diverse set of descriptions all of which mention skeletons, skulls, or "death's heads".

A "Death's Head" generally is depicted as a skull, so the specific mention of a crossed set of bones is not necessarily part of it. The symbol itself goes back to at least Roman Times.

As /u/davidAOP, mentioned many (but not all) of the popularly depicted "Jolly Rogers" are actually accurate, and depict various culturally symbolic images of death, time running out (for the victim as well as the hunted pirates), and violent actions.

There are a whole variety of "pirate flags" that actual pirates used, but there is also the pop-culture pirate flag. Are you asking what flags pirates used, or how the "Jolly Roger" became popular as a shorthand for "piracy?"

The best person to answer this is probably U/davidAOP, by the way.

See my first comment, I just edited it and added more information relevant to question.