I would like to use an external USB keyboard with the Odys Next Tablet. Does Android OS 4 support USB host mode and will an USB external keyboard work with that tablet if it's connected with an OTG cable? (I recently tried to use the keyboard with an Android OS 2.3 device and it's not working for it)
USB Host Mode was introduced with Honeycomb (Android 3.1, see changelog. This clearly explains why it failed on your Gingerbread (2.3) device -- and also should answer your question whether it is supported by Ice Cream Sandwich (Android 4.0) and above: Yes, it should work. As ov1d1u explained in his answer, it will require a "USB OTG cable" (OTG stands for "On-the-Go"). Does fine on my Cat Stargate 2 btw :)
Yes, it is supported by ICS or higher Android versions. But, it's not easy to find an OTG cable on the market. If you're like me and don't like waiting for eBay to have it shipped to you, here's a youtube link to make one yourself.
Requires some experience