Welcome to LinuxQuestions.
First, don't bother looking for .exe files in Linux. That's a Micro$oft executable file. In Linux, if the file has executable commands, the file must have the executable bit set. (Open a console window - the command line - and run the command ls -l. You will see files with three sets of letters - rwx which stand for read - write - execute permissions. For a file to be executable the x bit must be set.)
Second, pinging your desktop to laptop doesn't prove much. You are essentially on a LAN (local area network). Try pinging outside, such as LinuxQuestions. Open a terminal (the icon at the bottom of the screen that looks like a monitor) and run this command: ping www.linuxquestions.org
. If you get returns showing times for the signal to travel, you are connected to the internet through your ISP. If you don't see such times, you aren't connected.
To stop the ping, press ctrl c.