Remote deszmtop

“For ARD do you need a dedicated server for it if you want to schedule items?”

Not really. Useful if you do though depending on what you want to achieve.

“Here’s my question . . . If I run something will it "render" it on the client machine?”

Yes. The instructions (or task) you’ve sent to the controlled workstation will continue on that workstation until they’re complete. See below.

“So if I disconnect the main machine that has ARD installed things will keep going on the client machine?” “Or will it drop off when the ARD installed machine drops off?”

ARD Admin says as much on page 167:

https://www.apple.com/euro/remotedesktop/pdf/ARD_Admin_Guide_v3.3.pdf

This seems to be a popular question lately — how to perform remote access via VPN or otherwise — and this part is not really specifically related to ARD. The remote access is the "from anywhere" part of your question, and that tends to be the part of this that involves some amount of configuration effort and troubleshooting to get that VPN connection to work. Once the VPN is configured and stable, then ssh or ARD or whatever other IP-based protocol can all operate over that link. For what you're after with the remote display, you can use the built-in Screen Sharing server, and can use an iOS client from the app store, or you can use the OS X client.