Does Logos have an existing tool which identifies and gives the count of the most used words in a particular book?
Not yet. We should get that ability when Word Lists and/or Vocabulary Lists are implemented. See the link below.
No it does not and I don't know how any of the tools Logos does provide could help you accomplish this.
You can search an analytical Greek lexicon for references to a particular Bible book, but that gives you far more information than you probably are interested in.
There may be a resource in Logos that directly gives this information, but I am not aware of it.
Although if you run a Passage Guide on an entire book, I think the Interesting Words section may be similar to what you desire. It is omitting common words though, like "the", and "Jesus".
Have you considered the Analysis view of the Morph search?
Yep! That's the way I do it when I'm searching for word frequency. Make sure you use the lemma search and not just the word search, though, because there can be big differences between the two types of searches.