You can't drain an auto without a pump, unless you run it with one oil line disconnected and keep on topping it up. Reasonably easy to do, but I have never done it and it makes a mess.

The correct way of checking transmission fluid level is with foot on brake, cycle through all selector positions starting from P and ending with P. Leave in each position for 2 or 3 seconds before moving on. When done, leave the car idling for 30 seconds to a minute and check the fluid level while still idling (pull stick, wipe off, push in and out to read).

The transmission has electronic valves for selecting gears. On old transmisisons with mechanical ball valves your problem would be related to dirt or wear in the ball valve assembly. I would think that something similar could affect the modern electronic valves. Your car also has an electronic lock up torque converter which locks up gradually for first gear and reverse but instantaniously for other gears (don't quote me on this, I don't have the book with me now). So perhaps there is a problem woth the torque converter lockup and reverse is selected hard. The dealer can read error codes from the transmission controller if the controller detected any. In any case, I think you need a dealer to look at it. It is 3 years old, perhaps covered by dealer's warranty?