OK, well without the RTS line, you
will have to do the manual reset as described.
I also tried using an arduino mega as uploader. Didn't work.
Frankly, if the cable does not work, I feel it is better to continue to figure out
why it won't work, and address
that, rather than trying all the other ways that it will not work.
The other thing you need to try is to swap the Rx and Tx lines on the FTDI cable - still with the manual reset.
By the way, Arduinos - including clones - almost always come with the "blink" program loaded. If it is not blinking before you first try to program it, you would have reason to suspect the chip does not have the bootloader programmed either, in which case you would need to use your 2560
as an ISP to load that, as per Nick Gammon's instructions.