Just announced MacOS Mojave: what level of support for Metal is required

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.

That does seem a little strange. As a starting point it may be worth resetting both your SMC and NVRAM. Instructions follow.

Resetting the SMC on your particular MBP

1. Shut down your computer
2. Keep the power cable plugged in
3. Press at the same time shiftoptioncontrol (on the left side of the keyboard) and the power button
4. Let go
5. Turn your computer back on with the power button.

Resetting the NVRAM on Late-2016 MacBook Pro models

For all Late-2016 MacBook Pro models (with and without Touch Bar) follow these steps:

1. Shut down your machine. Yes, a full shut down, not just logging out.
2. Press the power button and then immediately press the commandoptionpr keys.
3. Keep these keys pressed down for at least 20 seconds!
4. Let go of the keys and let your Mac reboot normally.

Note: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Once you've tried these, let us know how you go.