In the irish folk song "Whiskey in the Jar" the narrator once "produces his pistol and his rapier" as if they were standard equipment, but I thought that rapiers were made for dealing with plate armour suits and that those were largerly obsolete by the time pistols were common.
Rapiers were not introduced to penetrate plate armour, but as civilian weapons for duels and street fighting. Generally speaking they were not carried on the battlefield, but about town. A rapier, or any sword for that matter, is not able to penetrate plate armour - the way to defeat armour with a sword is to slide the point into the gaps in armour. However this requires much finer point control than you can achieve with one hand positioned at the end of a (very long) rapier blade, and ideally calls for a very stiff blade. The best way to achieve this point control is through half-swording, or gripping the blade halfway up the blade with one hand (hopefully gloved) and keeping the other hand on the grip. This is not a rapier fencing technique. There were acutely pointed swords made for forcing their way through the gaps in plate armour, but these were stiff swords known Estocs (also known as tucks), not rapiers. In addition, many 'normal' swords of the 14th and 15th centuries are rather pointed, to better get through gaps in armour.
Regarding the timelines of pistols, they do overlap with rapiers - pistols come into use by 1550 or so, and so are used for nearly all of the Rapier's lifetime as a weapon (roughly, Mid/late-16th century-1700, depending on the nation). They were not part of the standard armament for a gentleman about town, but since the song describes a highwayman. It could be that the song's lyrics simply describe two weapons associated with dash and bravado.
See this thread for more comments about swords and armour and links to related responses.
Probably not in the sense that the Irish song is understood, but the common weapons of cavalrymen, especially in Sweden during the 30 Years War, were the pistol and the 'rapier'. I put rapier in quotation marks since it's a very different sword than we usually picture for the term. The blade is much shorter and broader, and the protection for the hand isn't as extensive. The pistol was the other primary weapon of the armored cavalryman in the period; Gustavus Adolphus famously told his troops not to shoot until they could touch the barrel to the enemy's breastplate. Rhe caracole was a common technique, where riders would circle in front of an enemy formation, firing when they had a shot before riding off in the circle to reload.