Where does the BitDefender engine reside?

The files are in the C:\windows\Downloaded Program Files folder but you cannot see them in Windows Explorer. You need to use the command prompt or a tool like ExplorerXP to see them. But there are multiple files and you would have to know all there names. I do not have a full list to give you. There are also entries that will be seen in a HijackThis log for Bitdefender. Just looking at the HijackThis logs in messages in this forum will show you those entries.

 

I would suggest you try the option under IE again and then also manually delete their files and folders. In addition to putting files in the Downloaded Programs Files folder, they also create this folder:

C:\WINDOWS\BDOSCAN8

I would delete this folder too after uninstall if it still exists. Then I would remove the below entries (if found) from Bitdefender as seen in a HijackThis log:
O9 - Extra button: (no name) - {85d1f590-48f4-11d9-9669-0800200c9a66} - %windir%\bdoscandel.exe (file missing)
O9 - Extra 'Tools' menuitem: Uninstall BitDefender Online Scanner v8 - {85d1f590-48f4-11d9-9669-0800200c9a66} - %windir%\bdoscandel.exe (file missing)
O9 - Extra button: AIM - {AC9E2541-2814-11d5-BC6D-00B0D0A1DE45} - C:\PROGRA~1\AIM\aim.exe
O16 - DPF: {5D86DDB5-BDF9-441B-9E9E-D4730F4EE499} (BDSCANONLINE Control) - http://download.bitdefender.com/resources/scan8/oscan8.cab

Then I would REBOOT before downloading it again.

Note: Problems may not be with Bitdefender. It could be something else on your system that is causing a problem.