What does the "pivotal role" mean? In what circumstances could I use it?

Why does she say "the country of Tanzania" instead of just "Tanzania"?

This is just a stylistic choice. The speaker may have added "the country of Tanzania" to add more detail about what Tanzania is (a country), or perhaps to add balance to the sentence. Just like some might choose to say "the color blue" instead of just "blue," the speaker may have preferred to say "the country of Tanzania" instead. Ex: I like the color blue the most. Ex: I like blue the most. Ex: The country France is famous for its cheese and wine. Ex: France is famous for its cheese and wine.

Is there a "water heater" as opposed to "water cooler"? And is water cooler same as water purifier?

My guess is that Tom Hiddleston is talking about the large plastic water bottles that are connected to machines to dispense water, or the larger type of water bottles that hold a large volume of liquid. Electric water coolers dispense both cold and hot water, so they are usually just called water coolers. Water heaters would refer to the type of electrical equipment that heats water for homes, for use when washing.

Unlike other accents, I think I've seen many comedians are making fun of Texan accents in their show. Why is that?

I'm not 100% sure. However, I would say that the Texan and southern accent is one of the most fun and distinguished accents to do. There are also a lot of stereotypes and generalisations of Texans that are easy to make fun of. Here are a couple of examples: Ex: Everybody in Texas listens to country music. Ex: In Texas, chicken is a vegetable unless it's fried.

What's "binge" mean? Can it be used in other contexts than in watching videos?

"Binge" means to indulge in an activity (to spend a lot of time doing one single activity without stopping, or to do too much of an activity). You can binge many different activities including eating, drinking, and reading. Ex: He admits to having an occasional ice-cream binge. Ex: I binged the entire series in one day.

Why "have been" was used here? What does it mean?

"Have been" (present perfect continuous) is used here because she is describing an action that started in the past, and has continued to the present. Here, her family started staying home in the past, and they are still staying home in the present. Here are some examples of how to use "have been" to describe something that started in the past, and has continued to the present: Ex: I have been reading a great book. Ex: She has been doing her job for two years. Ex: He has been busy since he started college.