Hmm... ok, I need to know a bit more before I can help you; did you buy and download your game from Steam, or did you buy an actual, physical copy? Because I got a physical copy of another computer game once, and it would not function at all. I couldn't even get it to start. So if you have a physical copy, you could have gotten a dud that just won't work. And second, do you have any mods that affect animals? If you do, one of them could be glitched, or conflicting with another one that you have.
As for your other comment, no, the random alternate start mod only alters the beginning so you start in a random location anywhere is Skyrim, instead of starting in Helgen. You still need to go to Helgen to start the main questline, but that's the only thing it changes, the rest of the game is unaffected.
Secondly, no, this glitch is specific to the opening, I belive it's called a "scripting error." The game is programmed to follow a chain of events, and when that "chain" malfunctions, you get a script error. You can encounter other glitches like this in future cutscenes, but since there's no other quest cutscenes that involve wagons in the game, no, you won't find this specific error again if you manage to fix it.