a) Once the dingoo is booted, the serial port should work like any other serial port on linux. It is registered as /dev/ttyS0 and you should be able to change the baud rate etc. Most (if not all) of the filesystems have inittab respawning a logon to the serial port but you can make your own rootfs easily enough that does whatever you want with the serial port.
b) The USB ethernet gadget is seen by Dingux as an ethernet port. inetd is running with telnetd and ftpd servers, so if you want your app to use / be accessed via ethernet, the easiest thing might be to make the app accessable / run through telnet. Or you could add your app to inetd to bypass logging in through telnet, or replace inetd entirely, depending on how complicated you wanted to make it.
For the application you mentioned, I'd think the easiest route would be to just use the dingoo's serial port. If you modify the rootfs so that the serial port isn't configured as a terminal line, then you can do what you want with it. There will be debug info on there only during the boot process, after that it would be quiet.