As a verb, what does "engage" mean?

Does "A be after B" mean "A is chasing B"? What does it mean?

Yes, "after" in this context means to chase, to look for, or to try to get. So to have something or someone "after" someone in this case means that they are looking to find that person or trying to capture them. Ex: You'd better hide. The whole city is after you.

What does "been around" mean?

Good question. "Been around" can have a few different different meanings. However, in the context of this video, "been around" means 'has been in existence'. So the video is saying that "red lipstick" has been in existence for over 5000 years. Ex: I wonder how long have cell phones been around? Ex: Disco has been around since the seventies.

Is it okay to say ”arm wrestling” instead? Is ”arm wrestle” a more common expression?

Both the word "arm wrestle" and "arm wrestling" are correct. "Arm wrestle" is the simple present verb for the term, as well as being the acceptable noun. "Arm wrestled" would be the acceptable past tense, unless used in a question form with "did". "Arm wrestling", however is present participle verb and gerund to show the ongoing act. Therefore changing this statement to "arm wrestling" would sound a little bit strange. Ex: Why are they arm wrestling? Ex: Why did they arm wrestle? Ex: They arm wrestled? Ex: I will arm wrestle you and win.

What's "walk the walk, talk the talk" mean?

To "talk the talk and walk the walk" means to say what you're going to do, and actually do it. It's a commonly used idiom, usually used to indicate if someone does or doesn't do what they say or preach that they do. Ex: Jim talks the talk about recycling, but he doesn't walk the walk. => doesn't actually recycle Ex: She talks the talk and even walks the walk with her values.

Is the word "zillion" used commonly?

"Zillion" refers to an extremely large number of things. This term is commonly used when emphasizing a large amount. It is not common in conversational English because it does not describe an actual number.