If you google "java frogger" this is the first thing that comes up: http://www.reinhardtfamily.com/Paul/Frogger/
I think a game like Frogger is a great game to start out with.
An easy way to implement the game timer is with a Thread. You can have it sleep for, say, 1000 ms (1 second) and decrement a counter variable until it hits 0.
To disable the mouseDragged method, just leave its body blank.
For your game mechanics, I'd make some sort of collection (probably an ArrayList) of cars and logs, give each car/log their own direction variable (Enumerations are great for this) and speed. For collision detection, the game is simple enough that you can have all obstacles check to see if they hit the frog every time they move. I hope this helps.