Where do I find these key items?

[soulxhawk]

The key item to do the mafia race and the second owl stage?

[怎么都是队友的错]

What owl stage need is in chapter 3 and act 2, in the well.

[Aurora Latios]

To do Train Rush, you need the Hookshot Badge from Subcon Well, Act 2 of Subcon Forest.
As for Cheating the Race, you technically need the Time Stop Hat, which should be the last Hat you get.

[wifeburger]

You can find the required item for Chapter 2: Act 4 "Train Rush" in Subcon Forest, (Chapter 3: Act 2 "Subcon Well.")

You can find the hat required for Chapter 1: Act 10 "Cheating the Race" in Chapter 4, Alpine Skyline.

[soulxhawk]

Awesome, thank you.

[MadJazzer]

Thank you!