I FIXED IT! More or less. My game is a bit unstable and crashes randomly sometimes, but that might be due to the mods I have installed.
Here's what I did:
Downgrade GTA IV to 1.0.7.0
Install LCPDFR 1.1 manually, using all files EXCEPT the following:
aCompleteEditionHook.asi
AdvancedHook.dll
AdvancedHookInit.asi
Download "3. LCPD First Response 1.0d_2 Manual Install.zip" from the official LCPDFR download page here:
Click the download button and choose from the list
From the zip in Step 3, put these files into your GTA IV directory:
AdvancedHook.dll
AdvancedHookInit.asi
Try the game to see if it works. One more thing: instead of dsound.dll I use xliveless as my ASI loader.
My theory is that the "patched" compatibility version of Advanced Hook uses changed memory addresses for its functions, so LCPDFR (being out of date as it is) gets confused and has its keybindings broken. Also, now that I'm downgraded to 1.0.7.0, I might try playing with LCPDFR 1.0d but with the new Scripthook and Scripthookdotnet from 1.1.
That was really not helpful. Did you even read my reply?