Hello Teldar, I would recommend you trying a different HDMI cable. Using another memory stick could be a good idea also.
It's a known issue that if you connect the DN2820 to a monitor/tv that doesn't support 1024X768 you don't get a boot screen. It doesn't affect the operating system so once you have an OS on it you should be fine (other than getting into the BIOS.
I hope it won’t be necessary to replace the NUC, but in case you need to, these are the 2 ways you can contact our warranty team. http://www.intel.com/p/en_US/support/contactsupport or submit a web ticket requesting the information. https://servicerequestmgmt.intel.com/webticketui/emailpage.aspx?lang=en-us