Why don’t the Lannisters use their money as more of strategic advantage during the war of five kings? They hire Tyroshi sellswords and Vargo Hoat, so clearly they’re not opposed to using mercenaries. Nor is there a “code” against it in Westeros – Stannis hires Sallador Saan, and in TWOW he wants to hire the Golden Company. You’d think given his money Tywin would hire 10 or 20k sellswords and use them to fight Robb Stark. But we never hear about the Lannisters approaching the Golden Company etc.

They certainly did, but there’s a couple factors that prevented them from making full use of those resources in the phase of the War of Five Kings from the outbreak through to Blackwater:

1. They’re on the wrong side of the continent, and Stannis and his navy stands between them and the Narrow Sea. So getting mercenaries to their army, and for that matter getting envoys to the Free Cities, would be rather difficult. 
2. Stannis is competing with them for mercenaries. He’s hired around 3,000 already. And mercenaries are not a fully fungible product – mercenaries might be already under contract (the Golden Company was under contract with Myr, and given the then-impending war between Tyrosh and Lys, it’s quite likely a lot of companies had already taken up contracts with them), or be in some other region of Essos where it’s harder to contact them. As the WOIAF says, “when not employed by the three quarrelsome daughters, the sellswords oft seek to carve out conquests of their own.”
3. The Golden Company is not really a possibility. Ser Jason Lannister died in the War of Ninepenny Kings, and Tywin, Kevan, and Tygett all fought with distinction against the Golden Company. Moreover, the Reynes and Tarbecks both fought for the Black Dragon in the First Blackfyre Rebellion, although they seemed to have reconciled with the Targaryens by the time of the Peake Uprising. 
4. There’s a timing issue. At the outset of the conflict, Tywin wants to mobilize before everyone else to stack the odds in his favor – the same way he did in the Rains. That means attacking ASAP rather than waiting for mercenaries to go all the way around Westeros to join up with his army.

And then in the second phase, Tywin’s got the Tyrells he can call on, and a rather negative experience with mercenaries maiming his son.