In response to the types of firearms used in the series, this website is a good source: http://www.imfdb.org/index.php?title=Firefly#MP40
As to the bullet traveling in space, as long as the firearm has oxygen, gunpowder, and a primer to make a spark (to ignite the oxygen and powder), the bullet will travel. And as anybody knows with items traveling with space, unless the item traveling collides with something else (space trash, satellite, moon, asteroid, meteor, the atmosphere of a planet), then that item will travel at it's original speed forever.
The gun that Jayne shoots (Vera) is technically a shotgun, but treated as a rifle in the series. The rounds that are loaded into 'her' are standard rifle rounds, with a brass casing, bullet, gunpowder, and a primer.
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