Translation:Please write your name.
The て form is used to connect the verb to other parts of speech, like another verb or another sentence clause.
Lol, typical anime sentence: new student > please write your name on the board and then go sit in the back next to the window xD
The word for write sounds like かぃ in the audio, but when I click on the characters individually the production is very different. Does that character have a different production in this context, or is my audio messed up?
Look at the spelling 書いて （かいて）- kaite - the te form of kaku - that's why it sounds like kai because it is kai in this form. Also I think you got pronunciation mixed up with production.
Omg all this talk of forms, progressiveness and particles is making my brain leak out of my ears. I come to comments looking to understand things better, but I just end up even more confused! sob Why!? WHY!?
It's all verb conjugation that Duolingo isn't actually teaching you, just showing you that it happens. Japanese conjugates verbs for different usage just like any other language and frankly I personally find it easier to comprehend than other language conjugation... But I learned it in a high school Japanese 2 class, Duolingo doesn't really care about teaching the rules, just the phrases.