You're very close with the idea that it's related to litters, but it's not related to scratching, or at least that's not a major issue as most mammals are born with soft claws (if applicable) I guess to prevent damage to their neighbours and the mother.
Eyes only open at a certain stage of brain development, and how fast your brain develops depends heavily on how long you've had to grow before being kicked out.
Taking a dog as the first example. Puppies tend to come in large numbers so pregnancy can't last too long due to space issues, but it's going to be very well protected by the mother and the pack so it can be born underdeveloped.
Now if you take a wilderbeast - there's to my knowledge usually only one (maybe two) born, therefore there's room to grow. Also it needs to be up and running pretty darn quick otherwise it's breakfast for a lion, so needs a fully developed brain & sight.
OK, people might seem to break this rule, but in reality although our eyes are open, the vision is pretty hopeless for the first month or so. It's just that we were in there long enough to develop our brains to allow sight. This can be shown if you look at premature babies as they often have their eyes tightly shut.
So in essence it's down to what evolution thought was the best policy given the circumstances.
Why hares have their eyes open - erm, yes. :-) I don't know a huge amount about the biology of the hare, but since it's fully fured I'm guessing they might need or some relative of theirs has needed to be up and running quicker than average.