Did the real medieval world have the equivalent of GRRM’s sellsword companies? If so, what were the names of some famous ones? How were they used? How much did it cost to hire one?

Absolutely. While individual mercenaries were used for most of recorded history, you start getting organized companies of mercenaries during the 12th century called routiers (bands) operating throughout France, Italy, and the Holy Roman Empire. Usually, they were organized and known from their geographic origins – thus the Genoese crossbowmen at Crecy, the Gascon infantry at Poiters, and so on and so forth. These bands would form the model for the rise of the condottieri in Renaissance Italy. 

Some of the most famous Medieval Companies were the Grand Catalan Company of the 14th century, which saw service in Anatolia argainst the Turks, and in the Balkans against the Byzantines and the Avars, the 14th century Navarrese Company, which fought in Greece, and John Hawkswood’s 14th century White Company, which fought in Italy. 

They tended to be a mix of infantry and cavalry – the White Company had 3,500 cavalry and 2,500 infantry at its peak, the Grand Catalan Company had 1,500 knights and 4,000 Almogavar infantry – often with some form of specialized skill that made them distinct from local soldiery. The Genoese were known for their skilled crossbowmen, the English for their longbowmen, the Swiss for their disciplined pikemen, the Almogavar for extremely fast-moving skirmishers. 

In terms of cost, they could be ruinously expensive. For example, Sienna spent the equivalent of $700,000 a year on mercenaries during the 14th century, badly stressing its fiscal capabilities. 

Equally importantly, the free companies were known for “living off the land,” pillaging widely, and ransoming regions (i.e, forcing people to pay protection money to avoid being pilaged).