Is it common to use "buttoned up" like this? What is the meaning of it?

How can I use “either”?

Here, "either" is used as an adverb. You can use an adverb "either" in a negative sentence instead of "also" or "too". In this case, "either" always comes at the end of the sentence. Ex: Jane can’t do it and I can't do it, either. Ex: It was a really nice restaurant, and it wasn’t very expensive either. Ex: A: I won't do it. B: I won't, either.

What's "demanding" mean here? I only know of "demanding" as in supply and demand.

Here, "demanding" means that something requires a lot of effort, skill, time, or energy! So, in a similar way to how we might "demand" a product or item, there are tasks that "demand" and require our energy, time, skill, and so on. Ex: Being a doctor or a nurse is a very demanding job. You have to do late-night shifts and focus for lengthy amounts of time. Ex: Construction work is physically demanding, so all the workers have to be fit enough to do the job.

I think "have been" is used very commonly in English, but I'm not sure when I can use it.

That's true! It is commonly used! It's the present perfect continuous tense. You can use it when something began in the past and is still ongoing now. Such as a habit, something that happens regularly, or an activity like reading. The form is "Has/have + been + [present participle]." Ex: I've been going to swimming lessons every Monday for the past two months. Ex: She has been playing violin since she was a child.

When can I use the expression “sth is all about sth”?

You can use this type of phrase to describe what the main point or focus of something is. This type of phrase would be suitable when making a declaration or announcement rather than everyday conversation. Ex: Today's episode is all about trying new recipes. Ex: My presentation today is all about how you can get healthy.

Would there be a difference in meaning if "everywhere" was used instead of "anywhere"?

There would be a difference! "Anywhere" implies whatever place, perhaps not all of them, but any of them. "Everywhere" means every or each place, all of the places. So, with that in mind, "everywhere" would have a more definite expectation of where they may be going, whereas "anywhere" is like, 'of all the possible places, you'll come with me.' Ex: You've been following me around everywhere. => each place I've been Ex: I'll follow you anywhere. Wherever you go, I want to go. I don't mind where it is. Ex: We've been everywhere and haven't found one single doughnut shop.