From the tests you made we can eliminate a list of potential culprit candidates:

1. memory: hardware tests

2. internal disk: use in target mode but you might be tougher on this test: in target disk mode fire Disk Utility and perform some (ex.: 4) Verify disk one after the other.

3. OS: re-install by an Apple service center

Here are the remaining candidates:

1. motherboard
2. graphical cards

Receipe to kill other culprit candidates

If your problem is just due to a graphical chain problem, your system might be correctly booted, but you can't notice it. To check this probability, you will have to set your Mac so as to be able to remotely verify that it is alive or dead.

• Connect the bad Mac to a good one with an Ethernet cable,

• Boot the culprit Mac, even if it does just show you his blue screen,

• Open a Terminal on good Mac, and get the broadcasting IP address. The following command will display it:

      ifconfig en0
  

  behind the keyword broadcast (3rd line). You may also get it at first with a slightly more tricky command:

     ifconfig en0 | sed -n 's,^.*broadcast \(.*\)$,\1,p'
  

  Let's say that it is broacast_IP (most probably it will be 169.254.255.255).

• do a ping broadcast to see how many Mac are alive:

      ping broadcast_IP
  

  if this command reply you with 2 different IP addresses, then you can conclude that bad Mac is alive and its graphical chain is dead.

  On the other hand, if you only get a reply of your good Mac, then bad Mac is really dead, and the last culprit is the good (i.e. motherboard).

Does that memory act as the "VRAM" that Final Cut Pro needs?

The integrated Intel Iris graphics chips contain onboard memory that act as L4 cache this helps speed up access to the shared system RAM, but doesn't replace it as the backing store for graphics data.

Does this mean that the discrete graphics card has 256MB on its own, but can also receive more from the system memory, a la the integrated cards?

The discrete card has its own RAM, placed near to the GPU and connected to the GPU with a private buss. Additional RAM from the system memory cannot be used with discrete cards. Really, you wouldn't want to -- it would slow down its processing capabilities.

Lastly, how different would the performance be for Final Cut Pro (or similar apps) between the Iris Pro graphics and the discrete card? If someone wants to use Final Cut Pro, is the discrete card a necessity?

This is hard to answer definitely. It depends on what a user is trying to do in Final Cut Pro. Certainly the discrete should out perform the integrated graphics setup when the workload starts to turn more demanding but OpenCL performance is proving to be almost on par for the integrated cards with the discrete cards in the latest Retina MacBook Pros.

Here is an article that has some discrete vs. integrated FCP benchmarking information in it. It points to this other article that has a choice quote in it:

In the past we sneered at the integrated GPUs and their puny performance. Not any more. The Intel Iris and Iris Pro are every bit the match or master of discrete NVIDIA Mobile GPUs -- at least when it comes to OpenCL acceleration.

Where the discrete cards win is in OpenGL-based workloads.

Ideally you'd run your own benchmarks on the hardware with representative workloads. If this isn't feasible, you're stuck relying on the benchmarks done by others like those two sites above to make your decisions.