Your first post in this thread says: "Computer worked on on-board video with limited resolutions choices and one monitor." But now you're saying you didn't try onboard video for the current problem. 
???
The onboard video port is covered because you have an add-in video card. The onboard port is automatically disabled by installation of an add-in video card, so that's why Dell covered it. Since it's disabled that's probably why you got beeps when you connected a monitor to it. You can use that onboard video port only if there's no add-in video card. Will use onboard video port removing graphic card as you suggested. A lot of information I have no clue about it like to remove video card to be able to use onboard video.
When I tried to install Windows on 2nd HD first one was disconnected. However computer behaved absolutely identical - Windows load files and computer powered off. But was the "2nd HD" connected to SATA0 or was it connected to some other SATA port? It makes a difference. And did you disconnect all peripherals, including network cable, leaving only mouse, monitor and keyboard connected when you tried to install Windows? I hope you realize that you will lose any data stored on that 2nd HD when you install Windows on it. All drives regularly backup so I am not afraid of losing any information and I know about it, but thank you very much. All peripherals disconnected. So far I have only 1 USB card for WD My Book external drive which I reinstalled to another slot since new card needed more space due to large fan.
If you boot from the original hard drive, using the new video card, does it load Windows or does it shut off before getting to the desktop or shortly afterwards? We need to decide if this is happening because the power supply is dying or if there's some other problem, so try to keep things simple with as few variables as possible. 
When I boot from original HD it loads Windows start up screen (black background with flying window and sign Starting Windows), freezes for 1 or 1.5 min and shuts off. It dose not go to desktop.