I suppose the phrase "I don't get it" is a little casual. What would be a good alternative in a formal situation?

How can I use “Go-to” in sentence

"Go-to" means something that is commonly chosen over another. "Go-to" can apply to pretty much anything: food, people, places, etc. You can definitely use this with another word! Ex: My go-to food is tacos because they are easy, delicious and versatile. Ex: Her go-to person is her dad. He gives her great advice about life, fixing things, and issues. Ex: His go-to place is the dog park near his house.

Would it sound awkward if I say "stay in" instead?

It would, yes! They use "at" since they're talking about Andy's place as a general location. You could use "in" for countries, cities, inside a space, and vehicles. Ex: I'm going to be in Spain for my vacation. Ex: I'll see you at the restaurant! Ex: I didn't see the keys in the car. Ex: Were you at the mall?

Why does it use "the" in front of "wind and sky"?

The definite article "the" is used here to make the nouns "wind" and "sky" specific. While the noun "wind" is often not made plural, it can be countable. Therefore an article can be used in front of both "wind" and "sky", and the definite article "the" is used to make them more particular.

What does "handle" mean here?

"Handle" means to deal with something or to be responsible for something. In this video, she is saying that she will deal with trying to stop Gru and the minions. Ex: She said she will handle our taxes this year. Ex: I will only take on what I can handle. Ex: He is handling the news very well.

Does the word "execute" and "executive" is somewhat related?

When "executive" is placed in front of a title (i.e. direction, officer, etc.), it means this person has the authority to make decisions on behalf of their organization. Thus, "executive director" means that the person is in charge of running the organization. It is related to the base verb "to execute," which means to carry out, perform, or implement an action. Ex: Our manager is in charge of executing our promotion strategies. Ex: The executive officer of our company reports only to the Chairperson.