I will often download .exe files and soon decide I do not wish do download them (just want to make sure I don't get a virus). Is it possible that I could get a virus from these downloads before I even run the executable?
It is possible for the act of merely downloading a file and doing nothing with it to be a security vulnerability, but this is true of any file type (not just exe files), and attacks of this nature are much more difficult than simply tricking random people into opening an untrusted file. This kind of attack would be exploiting some error in your browser or download manager's programming to execute arbitrary code rather than running a malicious exe file directly. This would essentially be equivalent to getting a "drive-by-download" virus from visiting a website, except it would be exploiting the download engine rather than the browser scripting or rendering engine.
Executable (exe) files are mainly dangerous because they run a program on your computer. If you run a program you are giving it permission to do anything your user account has access to do, including installing other programs, reading or destroying data, modifying system configuration, etc. If you do not run that file then it does not execute and you have not given it permission to do anything