A version of Linux Lite was successfully installed on TY's eMMC device last year. Check out this document for more information: http://www.intel.com/support/motherboards/desktop/sb/CS-034885.htm. Hopefully it will help you sort out the issue.
Regarding your second question, you should check out http://www.thinlinx.com/product.html. This is a zero-client O/S that installs on the eMMC; it not only supports RDP sessions but also provides GFX acceleration for such protocol, delivering ~60fps (if the server can keep up). There's some additional information on http://www.intel.com/content/www/us/en/nuc/nuc-thin-client.html.
I've tried several distributions of linux on the DE3815. Linux Lite is one of the ones that works, and works well. One thing to keep in mind is that due to the size limitation you need to create a single partition to install to. If you let Linux create multiple partitions the installation will run out of room.
Hope this helps.