i suspect that you have a different version of the bootloaders installed on your board. maybe an older one.. unfortunately we can't distribute bootloaders at his point of time, so you need to check with whoever 'gave' you this board to check if you can update. The fact that 'getvar all' doesn't work is a sign that something is not correct.
the host side fastboot will use 'getvar max-download-size' to cut the file to flash into chunks. Using '-S 256M' forces the chunks to be of 256MB which seems to workaround the issue, and clearly you could flash now (at least the OE image).
the last error 'size too large' means that you are trying to write an image that doesn't fit in 'system' partition. Here again this is a sign that you are using a different bootloader version compared to all the versions I have seen so far.. the OE image which is really small was flashed properly probably because it is small enough.
Maybe you can flash the debian image in 'userdata' with "-S 256M', and it might go further.
About your last sentence, what are you doing after powering the board? You need to compile and run your own kernel. are you doing that properly?
Also let me emphasize that using Linux on DB820c without the serial console is really not recommended, the state of the s/w is such that you will have to get a serial console. I would recommend to get the necessary equipment for that, or we are going to waste a lot of time.