How can I use a custom resolution?

It turns out that the resolution fix I mentioned in the question does actually work, but due to the new launchers provided in the Origin release of The First Decade you're unable to launch the game from the resolution fixer. What it does do though is make the required changes to the Red Alert 2 settings files (for both Red Alert 2 and the expansion pack; Yuri's Revenge) and these will take effect the next time you run the game through their official launcher.

I still haven't been able to get the game to run in windowed mode or prevent the game dying horrifically if you ALT+TAB out of it, regardless of compatibility settings or resolution settings chosen (including changing my desktop resolution to match the in game resolution, or changing desktop colour depth).

The changes that the resolution fix application makes are as follows, allowing you to replicate the settings without using it -

In your Red Alert 2 installation folder (in my case D:\Origin\games\Command and Conquer Red Alert II) you will find RA2.ini. If you have the Yuri's Revenge expansion as well then you should also have a RA2md.ini file in this location.

In these files you should have a [Video] section, which may or may not have values for ScreenHeight and ScreenWidth, if these do not exist you will need to add them but if they do already exist you will need to change them as follows;

[Video]
ScreenHeight=1080
ScreenWidth=1920

Using the above settings, the game will run at a resolution of 1920x1080 - many of the menu graphics do not stretch so they will appear with a large black border but once in game this will run as expected.