How many men do the brave companions actually have? And how does such a seemingly large multinational company come together, and where did Tywin find these evil fuckers?

They’re not that big, around 300 men. They’re a pretty recently-founded company, although they predated Vargo Hoat. 

As for where Tywin found them…although it’s not stated particularly clearly, Tywin clearly put the word out for mercenaries when he called the banners at Casterly Rock. 

Tywin didn’t seem to get very good/many mercenaries, for various reasons. My theory is that Tywin has a bad reputation among mercenaries for using them as arrow fodder to keep the wages bill down.

Among those he did get, the Bloody Mummers are in Tywin’s army after the Green Fork but we don’t hear of them before that although logically that means they would have had to be at the Mummer’s Ford and Tywin’s rampage across the hills of the southern Riverlands.