Do any cultures actually eat 'chilled monkey brains' like they show in 'Indiana Jones and the Temple of Doom' or was that made up by Hollywood?

Yes, it is mostly in China, but can be found elsewhere as well.
https://en.wikipedia.org/wiki/Monkey_brain

Served chilled or at room temp?

Monkey brain is a controversial delicacy, often attributed to the Chinese, but found in countries around the world. The practice of eating monkey brains has allegedly led to over-hunting, especially due to the belief that eating the monkeys' brain can cure impotence.

It is not only humans who eat the brains of monkeys. Two species of chimpanzee are known to eat the brains of monkeys to obtain fat in their diet.

https://en.wikipedia.org/wiki/Monkey_brain_%28cuisine%29