How is this is considered to be an Objectivist comic? Two chapters of the comic directly contradict Ayn Rand's views on war. Better Days has a strongly pro-war slant, in both cases glorifying military action in Vietnam and the Gulf War, whereas Rand said that when military action for any reason other than self-defense should be avoided, as self-defense is the only reason why it would be in the self-interest of a country to participate in war. Rand considered the United States' participation in The Vietnam War to have been borne out of altruism, which she opposed, whereas the Gulf War derived from similar reasons that did not involve the self-defense or self-interest of the country. In fact, the whole of Jay Naylor's oeuvre seems to suggest more of a conservative, if anything, leaning, rather than Objectivism, as none of the few elements that could be attributable to Objectivism also exist within Objectivism, and there's a pretty huge contradiction that separates this comic from being influenced by Randian philosophy. (There's also the anti-abortion stance in the comic, which differs from Rand's own views, as she was in favor of abortion.)
Some of what you cite happened before Jay's conversion, and also people are hypocrites about what they claim to believe sometimes, it happens.
Why are some canine character's tails cropped?
It was speculated on the Fridge Horror section for the sequel webcomic that cropped tails and ears are the equivalents of piercings.
The more likely answer is that Naylor has never seen those breeds in their natural state and thinks that's how they actually look.