This one was very unsatifsying. I don't mind stories which are all world creation and little plot (for example the previous story, Sugar), if the world creation is compelling enough. But this one - well, there were a lot of interesting ideas about this world - especially the idea of time travel as a class privelage - but the world never gelled together in a cohesive manner. And the disappearing animals was just to big a deal to just introduce as one more fact about the world. How distant a future was this? Not too distant, since their references were relatively recent (no-one was returning to the 25th century, for example). Now, over 500 years, a species could easily go extinct if it had a limited habitat to begin with, or if it was actively hunted or exterminated, or due to widespread environmental changes. But mice are way too spread out to be affected by any single event that wasn't deliberately aimed against them. And domesticated animals, well, they would have had people actively working to maintain them. I'm not saying that it's not possible to craft up a clever enough explanation. I'm saying that bringing it up as a plot point than failing to provide either explanation or any direct relevance to the main thread felt like the author was just throwing things in.
Also, I fail to see how killing a wild dog is in any way a bigger alteration than getting killed by a wild dog. Presumably, in the original timeline, the dog would have been prowling for some other prey. Now, it won't. It will either eat some of the dead timetraveller, and not only be full when it should be hungry (and thus maybe will avoid killing something else that should have died), but it will almost certainly bear viruses and bacteria unfamiliar to its environment. Or, it will be scared away by the timetraveller's entourage, in which case, it will probably end up somewhere different than where it should have been. Maybe it will now attack a child that would have otherwise grown up to be someone important.
Again, all of these issues are forgivable if they are used as part of an interesting story, or an interesting world. Here, the problems just accumulate and the payoff is missing.