How can I back-up my OS X instance and Restore from mounted drive? Synopsis

I have been having trouble With my MacBook Pro (Mid '12, 15")  I deployed tonnes of work Sunday evening/early hours Monday morning before I went to bed, leaving my MacBook running but locking screen.

Monday evening I returned home to do my personal weekly finance reports and start some work, where I clicked my mouse to get the screen on but it was white and nothing worked.  Simply a colour-wheel spinning infinitely.


I held the power button for a hard shutdown and turned it back on
The system booted up normally
I logged in and shortly after my MacBook started up all my desktop applications 
When I pressed ⌘ + _ the search was slow and opening applications took minutes.
Click on the close button on any application lagged, with at least 3 minutes pause
My mouse cursor moved, the display was frozen and I couldn't do anything.


Debugging Steps

I decided to investigate, after the 3rd time of clean rebooting after shutting down my applications using Firefox instead of chrome, closing all of my start-up applications, I un-installed Flash too.


The Disk Utility application ran tests on my partition and my disk reporting everything was OK (I then read this is not a reliable test)
I rebooted my Mac and using ⌘ + ⌥ + P + R and ran extensive tests on the RAM, it took about 2 hours, everything reported OK and passed (no failures).
I attempted to run further memory tests from Apple over the net using ⌘ + shift + D again, everything passed.


I thought it's the memory and decided to remove a stick of RAM and test if it failed, it did, I left 1 stick of ram out running on 8gb from 1 crucial stick. The same problems occurred as above.

I therefore assume the problem is my Hard Disk, regardless, I still want to back-up my installation as an image which can be restored from a USB storage device.

Question


Is it possible to boot-up my Mac from a USB external storage device and  install my existing OS as it is on a new hard disk?
I have read it's possible for Disk Utility to clone  my partitions and disks as a .dmg to an external drive - but how does this work with FileVault/encrypted disks?
Yes, you can do this with disk utility.  When you try to access the FileVault encrypted disk it will prompt for a password. You can then select it as the source for a restore to another drive.