Is my MOBO damaged?

Sorry to hear that, it could be any number of things.

Are you using a video card, or the graphics from the CPU?

If you use a video card, it is possible it is not seated in the PCIe slot correctly after the fall. Regardless of how it looks to you, remove it from the PCIe slot, check it for damage, and insert it again carefully. Remove and reconnect any power cables used by the video card.

The same thing goes for your sound card, which I assume is a separate card you added to the mother board? If not and you are using the speaker connections on the board's IO panel, that could indicate the board itself was damaged.

Be sure your DRAM memory is seated in the slots correctly and completely. You better remove them and insert them again to remove that from the list of unknowns.

If you are using the stock Intel CPU cooler, or even expensive CPU coolers, that may have moved from its mounting position on the CPU. That may be difficult to see. If you can get the PC to send a video signal to a monitor, you could either go into the BIOS and go into the Hardware Monitoring screen, or let it boot to Windows, and start a hardware monitoring program, like your board's F-Stream utility. What you want to do is check the CPU temperature. Just let the PC sit in the BIOS with that screen, or start F-Stream in Windows and watch the CPU temperature. If the CPU temperature keeps going up from 40C, to 50C, and to 60C while doing nothing else on the PC or on that BIOS screen, that should indicate the CPU cooler is no longer seated correctly.

If you see the mother board temperature rising, then the heat sink on the Z68 chipset may be loose. The large black heat sink around most of the CPU socket area, and also connected to the chipset heat sink might have moved out of place from the fall.

Regarding the idea about a fracture in the board and a short circuit, that is a possibility. But if it was a major crack in the board creating a short circuit, that would not allow the board to start at all. When you pressed the power button, it would run for a second or two, and shutoff quickly. You said the PC worked for about a week after being dropped, and still does not shut off until a few minutes have passed. That tends to indicate you don't have a major short circuit.

If you are using a standard HDD for the Windows drive, that could have been damaged by the drop. The power supply might also have some damage from being dropped, such as a component now being loose, or an internal connection being bad.

Your board has a Dr Debug LED display on the lower right side of the board. When you start the PC now, do you see a number displayed on it? You may see many numbers displayed and then change, but does it stop on one number?

You really need to strip down the PC to nothing but the power supply, board, memory, CPU and CPU cooler, with the monitor connected to the Intel graphics from the CPU, using an output on the board's IO panel. No drives connected at first, just pull the power cable from the drives. See if you can get into the BIOS and then do the CPU temperature test.

If that test passes, start to add parts to the PC ONE by ONE. First the drives, and then any video or sound cards. If something finally seems to fail, you probably found the damaged part.

Don't forget the power supply may be damaged. If you have another one you can use just for testing, do that as soon as possible.