The original solution to this error is located here thanks to protoman1008. However, the solution he provided seems a bit complicated for some people, and the .reg files that were provided in the thread are no longer available.
I have developed a little tool for people to use that greatly simplifies and automates the process of fixing this error. Pretty much a simple copy-paste, then a button press is all that's needed.
Before proceeding, please make sure that you are using an Administrator-level account on your PC, as Administrator access is needed for modifying the registry.
This is only for those who have encountered the "Registry is corrupt and has been altered" error when playing Red Alert 3 on Steam or Origin. DO NOT USE THIS TOOL IF YOU DO NOT GET THIS ERROR.
If you have any questions regarding my tool, please post them on the ModDB page for it! NOT in this thread!!!
Now then, why does this error occur? Sometimes, on installation of the game, the installer fails to write the game's "ERGC" key to the registry, which contains one's CD Key for their game. When the game tries to connect online, it checks if this key exists and if it's valid or not. Since the key fails to be written, the above error occurs.
To fix this issue, follow the below steps:
>>>> Step 1 <<<<
Download my RA3 Corrupt Registry Fixer. Unzip the .7z file you download (using 7zip or an equivalent) and launch the included executable as administrator.
>>>> Step 2 <<<<
Copy your Red Alert 3 CD Key to your clipboard.
1) Open your Steam library and find RA3.
2) Right click your game and hover over the "Manage" section in the context menu that appears.
3) Click the "CD keys" selection that appears when hovering over "Manage"
4) In the window that appears, click the "Copy Key to Clipboard" option.
1) Open your Origin game library and find RA3.
2) Right click the portrait for RA3 and click "Game Properties".
3) Select and copy your CD key in the window that appears.
>>>> Step 3 <<<<
Paste your copied CD Key into the small text box in the RA3 Corrupt Registry Fixer's window.
Press OK afterwards.
>>>> Step 4 <<<<
Re-hook your game to the C&C:Online launcher and try to log in again. You should be all set!