1. On my computer (Windows XP), Apoplexy opens a command line window before its main window, and it stays there while the program is running.
On your screenshot
I see that even this is missing, even though apoplexy.exe is running.
2. Why does it show "..." after apoplexy.exe? (In the "Имя образа" column of the task manager.)
Would there be something after it?
3. Also, why are there so many apoplexy.exe items in the list?
Did you try to start it that many times?
4. Can you start the DOSBox and ZSNES that come with Apoplexy?
5. You could try the WhatIsHang
First start this, then try to start Apoplexy.
If Apoplexy appears(*) in the WhatIsHang window, click on it and press F9.
Then copy the whole text into a text file and post it here.
(*) But I'm afraid it won't appear, because WhatIsHang lists only windows, but Apoplexy did not open any windows on your screenshot.