To begin with, you would not usually see a HDMI port or anything similar in device manager, you may see a disconnected device in the graphics driver control program. As you said, I believe this is simply a driver issue.
As for the shut down issue, I would also say it is overheating.
As for solutions, unless you are trying to watch blu-ray/HDCP protected content or are trying to view a very large resolution, there is no benefit to using HDMI over VGA. However, if you do want it to work, I would recommend you try the manufacturer supplied drivers (can be a pain, but should be tested for compatibility).
If you are still having no luck, you may want to double check in the BIOS that the HDMI socket is not disabled as it is sometimes possible to disable there.
Regarding the shut down, I would firstly launch Nirsoft Blue Screen View to double check it isn't crashing (so fast you wouldn't see it)... If it is and it is graphics related, this could explain everything - where as, if it isn't, I would think that your machine is simply overheating.
All I can suggest is to install a utility such as Spedfan, and take a look at the temperatures whilst playing a movie. If it is going insanely hot, make sure you are using your laptop on a flat even surface (such as a table - not a pillow/soft material!) and that none of the air intake/exhaust holes are covered.
If however it is not overheating, and the drivers do not fix anything... you could just have a faulty unit.
This will depend on the motherboard/BIOS. Some allow the onboard graphics others don't.
It appears that the Asus doesn't and to run a second monitor you'll have to use the 2nd output on the graphics card.
You don't have to buy a DVI lead - though using one will give you a better image - you can get DVI to VGA converters and use your existing lead.
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If you use the USB adapter, there would be no way to get the video to display on the laptop's built in screen. It would work with an external monitor though.