Q: Can I have the same wordpress site under different url? Can I run the save a wordpress site under different domain names? Like mydomain.com and mydomain.net ? Where the domain name will NOT change in the address bar of web browser.

A: Typically this is a really bad idea ( Google hates duplicate content).
If you need to do this you can edit your servers host file, but in reality unless you have a specific reason you should be using a 301 re-direct.
