How to program ATmega2560 chip
Yes, you can use an Arduino to program it, like any other ATMega chip.  You can also use a USBasp.  I suggest heading over to the Arduino website / forum / playground  where there are detailed instructions.

The chip is designed to be directly soldered to a circuit board using solder paste and a reflow oven or a reflow "rework" workstation.  There are QFN to DIP adaptors available - check eBay - but designing your own PCB and getting it manufactured is pretty cheap (and a good learning experience).