Does the Uber driver app show the destination of the customers before you accept?

Both don't show destination before accepting the ride. Uber and Lyft are testing to send you direction and duration if the driver is a slave and has 90% acceptance rates. Lyft will show destination when you arrive at pickup on the menu screen regardless of acceptance rate but then it's too late to cancel. 

We're true independent contractors all right.

No. All we see is how many miles, minutes to your location, and your rating. Once we accept we see address to your location that is all. However if your trip is more than 40 min we will see that its a long trip but not to where. I know my market well so I can usually tell if its a long trip where you are going depending on time of day or night. Although this is a guess even though I am right most of the time but this isnt a given in every market. Why do you ask, just curious:)

In general the answer is "no".  However there are some privileged drivers for both Uber and Lyft who can see the destinations before accepting. These are drivers who are highly rated, have high acceptance rate or have an elevated status, such as Uber's Pro Diamond.

No, but I wish they did.

No.  They do this to prevent drivers from cherry picking.