There are many programs to "backup" your eeprom. personally, I have always used Evox to do this.
in the system part of Evox, there is a "backup" button. click that, it'll say "backing up" then when its done there will be a new folder inside of your evox folder called backup. his folder contains your eeprom.bin, and hdd.txt which is what you need to backup your XBOX.
And as far as making your XBOX Linux only... cant...without a chip that is.
you can redirect your softmod, to have it launch straight into Linux after the softmod boot sequence. but you still need all your xbox files on C and E to boot. (without a chip)
your best bet is probably to use unleashX, and go to settings, and set the button autolaunch for Linux, if I hold down the white button while booting, I go to Linux, If I hold down Y I go to XBMC, If I hold down B I go to Avalaunch,etc...