@Hans83 I believe you're referring to the review from May that says:
"Sweet place, make sure you ask about pool, lake, gym access if that is what you are looking to do."
To answer your questions: no, and no. Nothing in this review violates Airbnb's content policy, and even if you can prove that the guest lied at some point in your correspondence, this has absolutely no bearing on the validity of the public review. And no, Airbnb does not change the star ratings just because the sub-categories are rated differently; guests enter an "Overall" rating separately from the categories, and it doesn't matter one bit what the "average" of the categories is.
Instead of wasting your time trying to get the review removed, perhaps you should ask to remove your Public Response, because what you wrote is a disaster. You come across as unprofessional and vindictive, drawing prospective guests' attention to the negative when left to its own devices, that review would scan as positive. It's shocking (in a very bad way) that you claim the guest "lied" about giving you 4 stars; why on earth would you put this kind of comment in a place that is visible on your own listing?
If you felt the comment about pool/lake access was misleading, all you needed to do was clarify what the cost and procedure involved is so that future guests have correct expectations. The fact that you instead chose to air out a payment dispute and attack the guest's star rating simply boggles the mind, and dramatically decreases the odds that an attentive guest will choose to book this listing.