The statistics associated with your userid show how many puzzles you started solving, and how many you actually solved.
When you push the New Game link, you are just being presented a candidate for starting and solving. It doesn't count against you to browse the available puzzles.
The information in the "This Puzzle's Statistics" box at the left shows the difficulty of the puzzle you will start if you push the "Start this Puzzle!" link.
There is no way to specify a certain puzzle difficulty, but if you hit the New Game link repeatedly, you will find a puzzle that suits your preference soon enough.