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"This 16-inch portable LED monitor is a must-have laptop accessory and receives both power and signal via a single USB cable for plug-and-play connection. Powered solely by USB, the monitor does not require a power cord or VGA cable." - From the First Link provided.

As you can see USB is not on the list of output connectors. Before portable USB monitors the only option was to buy a USB to HDMI/VGA/... adaptor. Using USB to output to a display device is relatively new and while Windows and Mac have drivers that support these USB monitors, Linux has not quite caught up yet.

[Second Part of Question 2]
On the first start the BIOS will probably not know what to do with the device. But that doesn't mean all is lost. This is a blog post by JoKi who bought the AOC E1649FWU (the monitor in your first link) and got it working with Ubuntu 13.04. But there are some posts in the comments section from people who followed JoKi's steps and couldn't get their portable USB monitor to work.

"A graphics processing unit (GPU), also occasionally called visual processing unit (VPU), is a specialized electronic circuit designed to rapidly manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display." - Wikipedia GPU Entry

"[A video card] is an expansion card which generates a feed of output images to a display , such as a computer monitor." - Wikipedia Video Card Entry

Buying a computer without any (as in zero) graphics capabilities and trying to output to a display device is impossible, unless your display device is a printer of some sort with a tangible medium. But for the questions sake, it all depends on the drivers installed on your system. Take for instance the output from syslogd on JoKi's Blog Post.

• How to Use an External USB DisplayLink Monitor as Secondary Monitor with Linux
• Using AOC USB monitor on Ubuntu
• Using DisplayLink USB Graphics On Ubuntu 12.04

_{[2] Difference Between VGA and Video Card/Graphics Card
[3] What is a video card?}

"This 16-inch portable LED monitor is a must-have laptop accessory and receives both power and signal via a single USB cable for plug-and-play connection. Powered solely by USB, the monitor does not require a power cord or VGA cable." - From the First Link provided.

As you can see USB is not on the list of output connectors. Before portable USB monitors the only option was to buy a USB to HDMI/VGA/... adaptor. Using USB to output to a display device is relatively new and while Windows and Mac have drivers that support these USB monitors, Linux has not quite caught up yet.

[Second Part of Question 2]
On the first start the BIOS will probably not know what to do with the device. But that doesn't mean all is lost. This is a blog post by JoKi who bought the AOC E1649FWU (the monitor in your first link) and got it working with Ubuntu 13.04. But there are some posts in the comments section from people who followed JoKi's steps and couldn't get their portable USB monitor to work.

"A graphics processing unit (GPU), also occasionally called visual processing unit (VPU), is a specialized electronic circuit designed to rapidly manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display." - Wikipedia GPU Entry

"[A video card] is an expansion card which generates a feed of output images to a display , such as a computer monitor." - Wikipedia Video Card Entry

_{(source: ddmcdn.com)}

Buying a computer without any (as in zero) graphics capabilities and trying to output to a display device is impossible, unless your display device is a printer of some sort with a tangible medium. But for the questions sake, it all depends on the drivers installed on your system. Take for instance the output from syslogd on JoKi's Blog Post.

• How to Use an External USB DisplayLink Monitor as Secondary Monitor with Linux
• Using AOC USB monitor on Ubuntu
• Using DisplayLink USB Graphics On Ubuntu 12.04

_{[2] Difference Between VGA and Video Card/Graphics Card
[3] What is a video card?}

• How to Use an External USB DisplayLink Monitor as Secondary Monitor with Linux
• Using AOC USB monitor on Ubuntu
• Using DisplayLink USB Graphics On Ubuntu 12.04

• How to Use an External USB DisplayLink Monitor as Secondary Monitor with Linux
• Using AOC USB monitor on Ubuntu
• Using DisplayLink USB Graphics On Ubuntu 12.04

In response to your first question, no a VGA card ^[1] is not required to display to a USB monitor. However, a USB card or port is required in order to display to a USB monitor.

[ Any PC that you buy will have a graphics card, either integrated and/or discrete. ] _[FIX-1] Integrated graphics are permanently embedded in the motherboard, while discrete graphics are actual cards that are plugged into the computer via AGP/PCI/... slots. Integrated graphics use system resources, eating into your RAM and CPU resources. The resources that discrete graphics require come on the card, so they don't put a heavy strain on system resources.

The following graphic shows the different types of output connectors that can be found on a graphics card.
Different types of output connectors. http://www.pchardware.co.uk/siteimages/Majority-of-VideoConnectors-vga-dvi-hdmi500.png
_{[Different types of output connectors]}

In response to your first question, no a VGA card ¹ is not required to display to a USB monitor. However, a USB card or port is required in order to display to a USB monitor.

[ Any PC that you buy will have a graphics card, either integrated and/or discrete. ] _[FIX-1] Integrated graphics are permanently embedded in the motherboard, while discrete graphics are actual cards that are plugged into the computer via AGP/PCI/... slots. Integrated graphics use system resources, eating into your RAM and CPU resources. The resources that discrete graphics require come on the card, so they don't put a heavy strain on system resources.

The following graphic shows the different types of output connectors that can be found on a graphics card.