Can you possibly deploy an application without the core from dev to testing? For example I do have an application with "Application" and "Application_Core", is it possible to only deply the "Application" alone? Thanks!

No,,,Your Application will still find the Application_Core which contain the Dependencies...

That depends actually. If you connect to your core through rest services, then you could potentially connect to the core of another environment. So then you wouldn't need to deploy your core, but you would still need to have it deployed somewhere.

Also, if you don't have any screens in your core application (which is best practice), then you wouldn't need to deploy it either. Because your core will be included in your main app as a dll. And all the references it needs will be deployed with it. (except for screens, services and maybe one or two other things)