I developed a SharePoint Site on Office 365 Developer Site.

I cannot open it in Visual Studio 2013 because I am getting the error: 

• "Remote SharePoint site connections are not supported." Office 365 API tools are installed.

I can however open it in SharePoint Designer 2013

I also tried the "Copy-SPSite" but received this error:

"The term 'Copy-SPSite' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again."

I am working with SharePoint 2013 which is required for Copy-SPSite.

I am new to SharePoint development, but am an experienced ASP.Net developer.... It just doesn't seem like it should be so obtuse to be able to do this.

What am I missing? 

Please check if the below links helps

https://social.msdn.microsoft.com/Forums/en-US/5983cf7d-c06a-4b38-a6cb-adef40f2bf9c/remote-sharepoint-site-connections-are-not-supported-error?forum=appsforsharepoint

Copy-SPSite doesn't work in SharePoint online only works in SharePoint on-premises.

Below are the list of commands that can be used in SharePoint online.

https://technet.microsoft.com/en-us/library/fp161364.aspx

In addition you can use CSOM and Powershell in SharePoint online

http://www.sharepointnutsandbolts.com/2013/12/Using-CSOM-in-PowerShell-scripts-with-Office365.html