No,bootloader does not go into EEPROM.
Sketch via bootloader - get an FTDI module accept USB from PC and output Rx, Tx, and DTR to your card. DTR will connect thru a 0.1uF cap to the uC's reset pin.
You can do the same an Uno - connect Uno's reset pin to Gnd. Connect D0 and D1 to D0 and D1 on the 2560.
Start a download in the IDE. When the IDE shows "compiled xxx of 130xxx bytes", something like that, press reset on your 2560 board. May take a few tries to get the timing right.
I am downloading Atmel Studio 6 to look at the bootloader placement question.