v3: Ok, I finally managed to run the game as it should, though still not using Wine directly, but it works.

Here's how I started it in Ubuntu and friends following vv221's previous advice:
1. Install PlayOnLinux
2. Download latest QuickTime for Windows (7.7.8 at the moment) from the official page
3. Open POL and in Tools > Manage Wine versions install 32-bit (x86) Wine 1.7.54
4. Open "Configure" from POL toolbar and create new virtual drive named "mojo" for example. In its General tab set Wine version to 1.7.54 (not System), then go to Wine tab, open Configure Wine and set Windows Version to Windows 7. Exit configuration window.
5. Run Install a program > Install a non-listed program > Edit or update existing application > check Show virtual drives and select mojo (if that's how you named it) > browse for downloaded QuickTime installation.
6. After QT is installed, start it and go to Edit > Preferences > QuickTime Preferences. Set both audio and video playback to SafeMode under Audio and Advanced tabs.
7. In the same manner install Bad Mojo Redux, just make sure to enter C:\badmojo as install location and when process is finished don't opt to launch the game but exit the installer. When POL asks you to make a shortcut, locate BADMOJO.EXE and select it, because you will use this executable to start the game directly.

Btw, this time there were no hook-related errors so no need to remove qthook.dll.

Sincere thanks to vv221 for helpful guidance and solution!

Glad to see your project is alive and kickin' :)