I've heard that the word "nerd" is considered rude. But, why would she call herself a "nerd"?

What does the "western" mean here? Does it refer to North America and Europe? Or simply Occident that contradicts Orient?

Good guess! In general, you can understand "Western" as meaning that which originates or relates to the West, which usually refers to the majority of Europe (Western, Northern, and Southern Europe), North America, and Australasia (Australia and New Zealand). Ex: There are often great cultural and ideological differences between the West and the rest of the world. Ex: Immigrants are often caught between Western values and the culture of their home country.

What's "straight" mean here? Is this a slang?

In informal terms, to "get something straight" means to get something right or properly. Here, "trying to get my money bag straight this time" means that he's trying to secure his income, to get his money without issues. And considering the context of the song, he is likely talking about a lot of money. Ex: Let's get things straight so we don't get confused later. Ex: I want to get my facts straight this time, so let me take some notes.

What does "see" mean and how can I use this expression?

"See" is being used as another way to say "I told you". It is commonly used when we are showing someone that we were right about something, or that we knew something would happen. "See" can also be used as a way to say, understand. Ex: See, I know she would be late. Ex: See? That's the right way to do it. Ex: See, he is her brother. Ex: Do you see what I mean?

How come "reason" became plural while "the only" came before it?

Great question! It became plural because the Prime Minister gave a few scenarios where it was okay for British people to leave their homes. If the Prime Minister only listed one reason why British people can leaven their homes, he would have said: "This is the only reason..." Ex: The only reason I am having issues with this homework is because it's math. Ex: I hate running. There are only two reasons why I would go running: something is chasing me or I'm in a rush to get somewhere.

Is "it is" skipped in this sentence? It seems like "Because it is critical thing..." sounds grammatically right.

You're right, it sounds like he did skip "it is" in this sentence. If he would have said "the critical thing we must do to stop the disease spreading between households is to stay at home" then it would have been grammatically correct. However, he left out the reason of staying at home at the end of the sentence so it is not grammatically correct and he should have used "it is" here.