Why are they even fighting anyway?

Swift is clearly upset or astound by the minimal differences that can lead to war, and separation. In the story, this is clearly represented with the heels. As I don't know much about English history I wouldnt be able to know the exact meaning of the "High-Heels" and the "Low-Heels", but I understand though, that he intends to say that their differences are ridiculous. Same thing happens with the eggs, and even with the conflict between Lilliput and Blefuscu which seems even more stupid.

Now, another small thing I want to talk about is the fact that Gulliver peed onto the pallace to save the emperor's wife. What does this mean? Also, the fact that he was granted pardon. What does this imply to society, or implied back then?

I'd say that first of all, Swift doesnt agree much with the royal desicions. Otherwise, why would he pee on their palace? He could believe that you ought to dishonor the existing rules, and crown, in order to save lifes. We have seen that Swift critizes the whole european culture, and that might've been another of his jokes. Aside from that, as Gullviver was pardoned, then he believes that there is entirely nothing wrong with doing so. That explains why he was so controversial with his book. (Anyways, I'm not sure if many people noticed back then.) Signifying nothing?