Is it okay to skip the noun after superlative? Shouldn't he say "one of my brightest and best STUDENT"?

What's the "rapid fire?" Is it like a "lightning round?"

It's similar, yes! "Rapid fire," in its literal sense, means to fire bullets quickly, one after the other. But we also use it to mean speaking unhesitantly and fast. "Lightning round" is mostly used within the contexts of games and contests, but it contains the idea of "rapid fire" to complete things quickly. Ex: We have some rapid-fire questions for you. Ready? Ex: The comedian had rapid-fire delivery. Ex: Okay, contestants, it's time for the lightning round!

Can I say "to touch" instead? As far as I know "touch" can be also a noun.

In general, yes you can say "to touch" because, as you said, "touch" can be a noun. However, in the context of this video, because the narrator is describing the action she's using the verb form. If you were to replace it with "to touch", the noun, this sentence would be grammatically incorrect. Ex: I want to touch the rabbit. Ex: She is touching the rabbit.

What's "made it" mean, and is it a common expression?

"Made it" means succeeded or arrived at something! It's a very common expression. You can use it when you've accomplished something difficult, when you're successful, or when you reach somewhere. Ex: Mark Zuckerberg has made it big time. => he's very successful Ex: We made it to the top of the mountain, finally. => arrived Ex: She kept auditioning, and, eventually, she made it.

What does "consigned to oblivion" mean? Is it a common phrase?

"Consign" means to give someone/something over to someone/something's care. "Oblivion" means a lack of awareness or consciousness. I believe "consigned to oblivion" means to be lost forever; no longer aware of something. In the address, he is hoping that the faults of his country will be gone. It's not a common phrase used in daily conversation but is usually used in funerals when someone dies, i.e., consigned (given over) to the earth.

How come does she use past tense here?

The party has already started, so MIllie Bobby Brown is asking why she did not see Cardi B at the party. For questions involving something that has already happened, or an incident in the past, the past tense is used. Ex: Why didn't you bring an umbrella today? Ex: Why were you late to school today?