Showing chronometer on PC controlling the begining and end of time by Arduino
It's perfectly possible to do, and the way I would probably do it is to use one of the ATMega32U4 based Arduinos (such as the Arduino Leonardo or the somewhat smaller Arduino Micro) to act like a keyboard. A keyboard with just one key - your chosen button.

Then you program the computer to respond to the keycode the Arduino sends to start and stop the chronometer. You could even use an off the shelf chronometer program that starts and stops with a specific keypress and tell the Arduino to send that keypress. It would probably take about 5 minutes to write the software - if that.