Does "allow me" is a more sophisticated form of saying "let me"?

What is the meaning of "nerve-wracking"?

"Nerve-wracking" is an adjective used to describe things or situations that are stressful or cause anxiety. The speaker has used to express his feelings being in the competition, as there is a lot of pressure placed on contestants right from the start. You can use "nerve-wracking" to describe similar situations. Ex: My exam starts in an hour. It is incredibly nerve-wracking. Ex: Giving a speech in front of other people is a very nerve-wracking experience.

Why is "do" used here? What does it mean? Can I skip "do"?

Here "do" is used to emphasize "know". Verbs are often preceded by "do", "does", and "did" to emphasize a particular verb, in this case 'know'. When you want to emphasize the main verb in conversation, pronounce "do" with emphasis. Ex： A: Do you like my new shirt? B: I do like your new shirt! Ex：So, you do want to talk about it?

I know "bit" is a past tense of "bite." But what does it mean here?

"Bit" in this case has a completely different meaning than "to bite". Here, "bit" means "part". It is common to use "bit" in this way in British English. Ex: You can throw that bit of carrot away, it's moldy. Ex: What was your favorite bit of the movie?

Can I say "local species" instead of "native species"?

Yes, you can. In this context, the two can be used interchangeably. Ex: This plant species is native to this region. Ex: This is a local plant species of the region. Ex: It is important to protect native animal species from being taken over by foreign species.

What is the meaning of the sentence “Time for your idea.”? Please rephrase this sentence.

"Time for your idea" here means "execute your plan now". The speaker is telling Dory that it is "time" to execute her "idea" (plan). "Time for your idea" is not a standard English phrase.