Those instructions were a little tricky, so here's what I tried:
First, I brought up C&C Online and hooked Kane's Wrath, then closed C&C Online. Then, I launched the game via the C&C3 And Kane's Wrath launcher via Origin. (the origin installation uses one launcher where you start it up and it asks you to choose which one to play).
Then, I clicked Kane's Wrath, which then took me to the C&C Online Launcher. At this point I went into Task Manager and found the task "CNC3Launcher.exe" (the only C&C Related process aside from C&C Online that had appeared in the Task Manager) and ended the process. Then, I went back to the launcher and clicked on Kane's Wrath. At this point, as in my original post, the C&C Online Launcher froze and a second instance of the C&C Online Launcher opened. This one behaved like I said--it would run the game but wouldn't connect to C&C Online.
Ending C&C3Launcher.exe before choosing either C&C3 or Kane's Wrath (before it takes me to the C&C Online launcher) simply shuts it down on the spot.
One thing I noticed while going through this is that the C&C3Launcher.exe process actually closed on its own if I waited a bit. This didn't affect whether the C&C Online launcher worked, however.
Update: While I was typing this I had both the frozen and the second C&C Online launchers still up. After a while I noticed that (apparently) the frozen one woke up and I could interact with it. I tried launching C&C Kane's Wrath via the one that was frozen, and it produced this error:
I closed this out, and then tried the other launcher (the one that opened when the original froze), and it gave me the same error.
I'm now working on backing up save files since I'm starting to think this inevitably means I have to re-install both of them.
EDIT: So I reinstalled both games. They are both fully vanilla--no mods, not even the fixed launchers. C&C Online will launch the game, but despite everything I have tried, I can't connect to Kane's Wrath or Red Alert 3 online now. I ran it as administrator and turned off my firewall to no avail. Yeesh.
EDIT #2. I tried installing the fixed launchers, and the issue came back. So this basically means I either can play online (which, technically, I can't still, due to the above issue), or I can use mods (and since Origin's installer doesn't differentiate between the two games, I can't, say, install the fixed launcher for C&C3 so it can run mods and leave Kane's Wrath as vanilla so I can play online.)
Unless you guys have any suggestions for me, the outlook seems bleak. I'm guessing the vanilla game breaking is due to my university's firewall being a jerk since it worked fine at my folk's place.