I think there could be quite legitimately (please pardon the pun) a separate forum on WAB for disputes with neighbours over such matters. It is dodgy ground (*sigh*; another unintended pun) and something that cannot usually be answered in a quick "yes" or "no" manner. It is a very subjective area, prone to what has preceded beforehand, what the reasonable person would consider, etc...and all this depends on location as well ("Disgusted of Tunbridge Wells" may be less tolerant than "Ms I inherited this ramshackle place from my ancient family", for instance.)
To me, the tree does not look unreasonable, but who knows, in a court of law with a clever advocate, the dispute for light, subsidence (due to roots) etc could be argued.
I am currently concerned about an Ash tree at the bottom of our garden. It is at the bottom of ours, yes, but as the bungalows behind us have handkerchief sized gardens that face South-easterly, I cannot see it being a popular addition to our garden! (Hence my ruthless pruning every year
.)
As for badgers, yes, they will head butt their way through almost anything. My neighbours are fighting a losing battle over this, lol! (Why don't they just give in and appreciate how lucky they are to have urban badgers???)
D.