I made a test with a pre-built image from banana-pi.org.
This is Debian 8 "Jessie" with a very old kernel 3.4.
I assume, the famous sunxi-kernel.
Here, the Ethernet works.
The complete log during booting is attached.
Hopefully, it helps.
To your questions:
> * Which power connector is used? ( barrel or microUSB )
> * What power supply is used? ( Volts, Amperes )
5V / 2,5A
> * Did tests happen with other cables? [y/n]
> * Did tests happen with other power supplies? [y/n]
No. If necessary, i can repeat the tests with my laboratory-power-supply.
If i can do further tests, or if you need further informations: please let me know.