I assume the word "detective" is used for police and private investigator. However, how do I distinguish them in plain text?

What does “walk right in” mean?

Good question! "Walk right in" means to walk directly into a room, place, or situation. Here, "right" means "directly," "straight," or "abruptly." Ex: I walked right into my manager's office and asked for a vacation. Ex: She walked right into my life and changed everything. Ex: He walked right in without knocking.

What's "do over" mean? Is this a common phrase?

"Do over" can be understood quite literally - it means to do something again, to try or attempt something again. Ex: I want a do over of the test. I didn't do very well. Ex: Can we have a do over? I promise to behave better next time.

Is there any difference in meaning between "should go" and "should get going"?

Yes, there is a difference in meaning between "should go" and "should get going". To "get going" means to start going or get ready to go, especially when there has been a delay or when someone is already late. Ex: We should get going, it's going to take awhile to get out of here. Ex: We got going around 7 but we didn't actually leave until 8.

Why is "They go" translated as "They say"?

"They go" is translated as "They say" because in this context, it means that these individuals are saying these particular things about her. It is also an artistic choice. The musician may have thought that "They go" sounds better than "They say" in this particular song. When in doubt on which one to use, stick with "They say."

Can you tell me how to use the word "gotta"?

Sure! "Gotta" is another way of saying "got to" or "have to." You can use "gotta" when you are telling someone that they are required to do something. Ex: You gotta tell him how you feel. Ex: We gotta go to the grocery store. Ex: He's gotta paper to write. He can't hang out with his friends.