The difference between chidori and rasengan is not their power difference, but their efficiency. Sasuke can only use 2 chodoris in one day, any more will kill him. Naruto can use rasengan over and over and over...and over without any damage to his body.
Except that Sasuke used Chidori I believe 4 times against Gaara that one day, and he lived (though he did have a chance to die on the 4th use, he still used it 3 times without the seal).

This MIGHT be a weakness for Chidori, but is the Sharingan required? I don't really think so since I think the Sharingan just helps make the target easier to connect with, not to mention dodge some attacks (Sasuke vs Gaara).
Yes, Sharingan is required to use the Chidori, as stated by Kakashi in one of the training flashbacks. Sasuke has to be able to predict the movement of his target, or else he cannot dodge any attack they might throw at him, since he puts all his power into running and straight-thrusting (as explained by Gai, and an example given by Rock Lee).

I know someone would bring up the hand thing...that doesn't count.
It quite counts. Without Kyuubi's regenerative powers, Naruto could burn his hands off using the technique too much. Remember, he's rotating a very condensed mass of air on his hand, you can't do that without taking some kind of damage after a while, even if it's contained.