Does that mean if I use a USB-to-TTL cable (Such as the 'USB to Serial TTL Cable' from CoolComponents, say) I can get rid of the usb chip and program the ATMega directly?
Yes. If you look at the older Mega schematic (not the new 2560 one) you can just remove the section that includes the FT232. The pins that would normally be connected to the FT232 would just need to be brought out to a header. You are basically just moving the FT232 to an external board. I think it is pretty easy once you study the schematic, but let us know if you have any questions.
Again, I know that for the Arduino 2009/Uno you can burn the bootloader via the USB if you use the Arduino IDE
I don't think this is possible with a standard setup. You can't just plug a standard Arduino in that has a virgin chip and load the bootloader. If this was possible than there would be no need for the bootloader to begin with. It is possible to modify the Arduino to self program by using the bit-banging method, but lets pretend that doesn't exist for the moment.
To program the bootloader into the Atmega1280/2560 you will need to use an ICSP or Parallel programmer. The ICSP is the easiest way. You can either bring out a 6 pin header (2x3) that is similar to the ICSP headers on the standard Arduinos or just connect a programmer to the correct pins. But either way you will need some separate programmer that is supported by avrdude and can program using ICSP. So you would first build your board then program the chip with the bootloader. Again the ICSP programming header configuration can be obtained by looking at the Mega schematic. Essentially you need to connect MISO, MOSI, CLK, RESET, GND, VIN