How to flash a RobotDyn UNO R3? (CH340/ATmega168PA 5V)

I have an RobotDyn UNO R3 and I can't seem to be able to flash it.
I can flash other CH340 based devices just fine using the Arduino IDE (1.8.5).
When I connect the RobotDyn UNO R3 it shows up as a serial port under tools->port.

I think I have successfully flashed such a board before in the past and the trick was to select "Arduino Leonardo" instead of "Arduino/Genuino Uno", but I've tried that and also several other options like "Arduino/Genuino Micro" and it just doesn't work.

I'm on Linux btw.
Port: "/dev/ttyUSB0"
Programmer: "AVRISP mkrll"

Here is the output (everything is set to verbose):

When I select "Arduino/Genuino Uno" under Tools->Board:

Sketch uses 3758 bytes (11%) of program storage space. Maximum is 32256 bytes.
Global variables use 209 bytes (10%) of dynamic memory, leaving 1839 bytes for local variables. Maximum is 2048 bytes.
/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/bin/avrdude -C/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf -v -V -patmega328p -carduino -P/dev/ttyUSB0 -b115200 -D -Uflash:w:/tmp/arduino_build_628016/Blink.ino.hex:i 

avrdude: Version 6.3, compiled on Jan 17 2017 at 11:00:16
         Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
         Copyright (c) 2007-2014 Joerg Wunsch

         System wide configuration file is "/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf"
         User configuration file is "/home/fedora/.avrduderc"
         User configuration file does not exist or is not a regular file, skipping

         Using Port                    : /dev/ttyUSB0
         Using Programmer              : arduino
         Overriding Baud Rate          : 115200
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 1 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 2 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 3 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 4 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 5 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 6 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 7 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 8 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 9 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 10 of 10: not in sync: resp=0x00

avrdude done.  Thank you.

Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions

When I select "Arduino Leonardo" under Tools->Board:

Sketch uses 6090 bytes (21%) of program storage space. Maximum is 28672 bytes.
Global variables use 172 bytes (6%) of dynamic memory, leaving 2388 bytes for local variables. Maximum is 2560 bytes.
Forcing reset using 1200bps open/close on port /dev/ttyUSB0
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
PORTS {/dev/ttyACM0, /dev/ttyUSB0, } / {/dev/ttyACM0, /dev/ttyUSB0, } => {}
Uploading using selected port: /dev/ttyUSB0
/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/bin/avrdude -C/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf -v -V -patmega32u4 -cavr109 -P/dev/ttyUSB0 -b57600 -D -Uflash:w:/tmp/arduino_build_628016/Blink.ino.hex:i 

avrdude: Version 6.3, compiled on Jan 17 2017 at 11:00:16
         Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
         Copyright (c) 2007-2014 Joerg Wunsch

         System wide configuration file is "/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf"
         User configuration file is "/home/fedora/.avrduderc"
         User configuration file does not exist or is not a regular file, skipping

         Using Port                    : /dev/ttyUSB0
         Using Programmer              : avr109
         Overriding Baud Rate          : 57600
         AVR Part                      : ATmega32U4
         Chip Erase delay              : 9000 us
         PAGEL                         : PD7
         BS2                           : PA0
         RESET disposition             : dedicated
         RETRY pulse                   : SCK
         serial program mode           : yes
         parallel program mode         : yes
         Timeout                       : 200
         StabDelay                     : 100
         CmdexeDelay                   : 25
         SyncLoops                     : 32
         ByteDelay                     : 0
         PollIndex                     : 3
         PollValue                     : 0x53
         Memory Detail                 :

                                  Block Poll               Page                       Polled
           Memory Type Mode Delay Size  Indx Paged  Size   Size #Pages MinW  MaxW   ReadBack
           ----------- ---- ----- ----- ---- ------ ------ ---- ------ ----- ----- ---------
           eeprom        65    20     4    0 no       1024    4      0  9000  9000 0x00 0x00
           flash         65     6   128    0 yes     32768  128    256  4500  4500 0x00 0x00
           lfuse          0     0     0    0 no          1    0      0  9000  9000 0x00 0x00
           hfuse          0     0     0    0 no          1    0      0  9000  9000 0x00 0x00
           efuse          0     0     0    0 no          1    0      0  9000  9000 0x00 0x00
           lock           0     0     0    0 no          1    0      0  9000  9000 0x00 0x00
           calibration    0     0     0    0 no          1    0      0     0     0 0x00 0x00
           signature      0     0     0    0 no          3    0      0     0     0 0x00 0x00

         Programmer Type : butterfly
         Description     : Atmel AppNote AVR109 Boot Loader

Connecting to programmer: .avrdude: butterfly_recv(): programmer is not responding

avrdude: butterfly_recv(): programmer is not responding
avrdude: butterfly_recv(): programmer is not responding
avrdude: butterfly_recv(): programmer is not responding
avrdude: butterfly_recv(): programmer is not responding
avrdude: butterfly_recv(): programmer is not responding
Found programmer: Id = ""; type =

And for selecting the "Arduino/Genuino Micro" board the output is pretty much identical to the "Arduino Leonardo" output.

Edit:
I've also tried holding the reset button of the board until the upload starts: I got the exact same errors.

What microcontroller does the board have? The Banggood page you've linked clearly says it has an ATMega328P, but your subject title says ATMega168PA. If you really do have the 168PA you'll need to select 'Arduino Duemilanova or Diecimila' as the board type, and the appropriate processor type.

felic:
I think I have successfully flashed such a board before in the past and the trick was to select "Arduino Leonardo" instead of "Arduino/Genuino Uno", but I've tried that and also several other options like "Arduino/Genuino Micro" and it just doesn't work.

That will never work, the Leonardo/Micro use a completely different microcontroller to the Uno.

BJHenry:
What microcontroller does the board have? The Banggood page you've linked clearly says it has an ATMega328P, but your subject title says ATMega168PA. If you really do have the 168PA you'll need to select 'Arduino Duemilanova or Diecimila' as the board type, and the appropriate processor type.

Oh, I'm sorry I didn't notice that. I replaced the link with another one now. My board definitely has an 168PA.
Unfortunately selecting "Arduino Duemilanova or Diecimila" resulted in this:

Sketch uses 3758 bytes (12%) of program storage space. Maximum is 30720 bytes.
Global variables use 209 bytes (10%) of dynamic memory, leaving 1839 bytes for local variables. Maximum is 2048 bytes.
/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/bin/avrdude -C/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf -v -V -patmega328p -carduino -P/dev/ttyUSB0 -b57600 -D -Uflash:w:/tmp/arduino_build_628016/Blink.ino.hex:i 

avrdude: Version 6.3, compiled on Jan 17 2017 at 11:00:16
         Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
         Copyright (c) 2007-2014 Joerg Wunsch

         System wide configuration file is "/home/fedora/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/etc/avrdude.conf"
         User configuration file is "/home/fedora/.avrduderc"
         User configuration file does not exist or is not a regular file, skipping

         Using Port                    : /dev/ttyUSB0
         Using Programmer              : arduino
         Overriding Baud Rate          : 57600
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 1 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 2 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 3 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 4 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 5 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 6 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 7 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 8 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 9 of 10: not in sync: resp=0x00
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 10 of 10: not in sync: resp=0x00

avrdude done.  Thank you.

Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions.

BJHenry:
That will never work, the Leonardo/Micro use a completely different microcontroller to the Uno.

Unless the clone simply uses a different chip.

Do you have another Arduino board on hand, or an ISP programmer? If so, I would recommend just burning the bootloader so you will know for sure which one it's using. Minicore has support for the ATmega328PA and provides the top of the line Optiboot bootloader: