What is the most obvious translation of verb "mitmachen" in this sentence? Isn't "to participate" more adequate meaning? Isn't "beitreten" more accurate word to describe "to join" here?

pronunciation is terrible.

Warum ist das risponte false? Why do you(formal) want to participate?

(another modal verb for "join" is "mitkommen." An example is: "Ihr kommt auch mit?" " are you all also joining?" )

Cant understand mans annunciation

OK I guess i need another explanation, why could it not be " why does she not want to join"?

I thought mochten was past tense. I thought it was "Why did they want to join?" Isn't mogen present tense?

Can I not use "how come" for "warum"? I was under the impression that "wieso" was "why" and "warum" was more like "how come"?

What is the differene between möchten and wollen?