It isn't necessary because it is clear that it is the book that will be given. However, the sentence will sound better with её. And to make it more clear one can say "и дам ее тебе почитать" (lend to you) or "и подарю ее тебе" (give it to you as a present) or "и верну ее тебе" (return it to you). The sentence as it is presented by DL can be extended like this "и дам тебе этой книгой по башке" (and hit you with that book on your stupid head) - this meaning can be easily read into it especially because the word найду was recorded with a distinctively threatening intonation.
The word "найду" in itself is not threatening, it is supposed to be a promise. It is the intonation in the recording that made it sound threatening. It has been a year since I heard it so it is hard for me to tell exactly why (I can't listen to it again as I don't know where to look for the recording). Unfortunately, the automated recording system used by DL often presents wrong intonation patterns. As a result, a statement may sound like a question and a promise may sound like a threat. I always report such cases.
All of this aside, in this instance, "the" is a better translation than "this". English speakers only say "I will find this <noun>" when they're feeling poetic (e.g. "I will find this man" as opposed to "I will find him"), because to English speakers, "this" refers to something right in front of them, which you generally don't need to go find. Generally, the phrase is "I will find the <noun>", (or "I will find that <noun>") and here I would argue that the это indicates a specific book ("I will find the book" as opposed to "I will find a book"), not "this" book.
?? Still confused. Are you saying that the "hints" are bi-directional, and also used with "English for Russian-speakers," and that "эту" can be the "hint" for "the"?
In this sentence, I felt like "the" made more sense. "This" sounded overly specific for a book that was lost or somehow unavailable. "The" made more sense, as in "I will find the book we've been talking about" or "I will find the book that you need."
"This book" would be a book that you're holding in your hand, or pointing to on a shelf, or pointing to in a list of titles, or some such.
The hints are indeed bi-directional.
In the course of English "the" is treated the same as "this" in most cases, so that's what the hint suggests. A native speaker of English, though, does not need to be constantly reminded that articles are not optional in English. So in this course, a and the are not translated anyhow. You are expected to come up with them on your own. When you see a form of этот, we expect "this" or "that" as a translation, depending on what makes sense.