One thing that comes to mind is rampant racism, with everyone so significantly different and the biological lines between humans and animals being blurred, there would be some serious accusations that one species is less intelligent than the other and vice versa. Because of this I feel that the world would be relatively divided and true globalization would be stunted, we'd basically go back to an extended dark age of war as countries form for species banding together. The division, I believe, would only be more or less a temporary thing, because there'd be too many species to hold a country for each single one, so species would band together to create unions, but racism within these unions could still persist. Basically with the increased variety between humans comes an increase in separation and conflict.
Depending on the extent of peoples transformation, technology would be substantially different, the species without opposable thumbs would find it difficult to hold things so different sorts of models would have to appease these species. Different species have different shapes and forms so more variety in technology would need to exist, this would mean that the advance of technology would be largely slowed down as less people would be working on the same technology OR technology would need to be universal to all species which would take significant longer. The increase in war would mean an increase in technological progress, but again the technology created would be separate and akin to each individual species.
Most current religions would probably be thrown out the window, but new religions may be created or the current religions adapted to fit with the phenomena. Regardless, the religions of this world would be substantially different from the religions of today. I would agree with Razot's idea, I don't believe people would go all vegetarian, but again with an increase of separation can come a larger diversity of ideals. So while some species may endorse a vegan diet, others will simply hunt for prey, this is what is better suited to the species and it creates a separation. It's hard to say if species with human-level intelligence would eat other species with human-level intelligence. It's likely this act would be committed at first, but as more unionism gets created this would be largely outlawed, provided there isn't some dominant species who has enslaved the rest.
So the world seems a bit more bleaker in general, but I believe general life-spans would rise because of the nature of the species themselves, being stronger, faster, and more able may make it easier to live and make the world less harsh