I don't understand the phrase "For if by some minute chance you meet the challenge," in this scene. Is it an old English, perhaps?

Is it grammatically right to say "put your VR goggles on"?

Yes. You can restructure the sentence as you have done without affecting the grammatical correctness of it. Both "please put on your VR goggles" and "please put your VR goggles phone on" are grammatically correct and have the same meaning.

Is "one certain patch of" necessary in this sentence?

Yes, "one certain patch of" is necessary here because it specifies that rubber was rubbed off of only part of the tires, therefore the tires became uneven and unstable on this car that did these stunts.

I've heard "check it out" a lot, but what it means?

To "check it out" or to "check out [something]" means to look at something and take notice of it. Ex: Let's check out the amusement park this weekend. Ex: Check it out! I have a new skateboard. => showing the skateboard to someone

Is it common to use the word "bleed" to non-human thing?

The verb "bleed" can refer to different liquids that seep out or flow from something and it doesn't just have to describe "blood". Thus, it can be used to describe liquid flowing from objects, however it is still most common to use in the context of a person or an animal that emits blood. Here's a few examples of "bleed" for non-human objects: Ex: The ink from the pen bleeds through the page. Ex: The sap is bleeding out of the tree.

What does "exempt" mean?

To be "exempt" from something means to be free from or an exception to something. The speaker has used this word to express that although Anne Hathaway's character is not interested in fashion, she is not free from (exempt) the influence of the fashion industry. Ex: The company has an exemption and does not need to pay taxes. Ex: You are not exempted from following government orders.