Can you dump the contents of your phone (os + apps + data) and run that in an emulator/Android Studio? Is there a way to dump the contents of your phone (OS + apps + data) and run them on a PC in an emulator of some sort?

I feel like there are security implications to being able to do this but with physical access to a device my experience has been that for the most part security simply disappears. I'm not knowledgeable about how custom ROMs etc are made but feel like there might be tools to do what I'm speaking of. 

Backstory: Basically, my phone's screen died and I am in the process of pulling contents off of the phone. I'm using AndroidScreencast to interact with the GUI. This is a slow process as the update of Android Screencast is a bit slow. I'm using adb shell to simply cat files to the sdcard.
This process made me wonder if there was an easier way.  Is there a way to use dd (or some other application) to pull the entire ROM and then emulate that such that I don't have to use the android screencast? 

The device in question is running an unrooted version of Android 2.3? (I don't have the device in front of me now so I can't post that.)  I have been able to get root access using exploid and that is how I've been sending files to the sdcard for retrieval.

To reiterate in a slightly different way, can you backup your os+apps+data via a dump and then run them locally on a pc somehow?
No, generally speaking there is no way to do this.  Your ROM is customized for your phone and is pretty tied to that specific hardware.

You could, of course, pull the apps and install them into an emulator, and then back up the data for them on the phone and restore that data on the emulator.  With root and 2.3, the simplest way to do this might be via an app like Titanium Backup.