if the build fails, then nothing happens. all that happens during the build is that you're creating a new file to take the place of the old kernel. the kernel that is currently working will not be affected until you either reboot or run lilo. most distros have the kernel image in the /boot/vmlinuz file. at boottime, the kernel is loaded from this file. from that point on, it resides completely in memory, so the contents of vmlinuz don't matter until the kernel is reloaded.
the instructions should be distro-independant. as for logging in, it's generally safer to do a kernel rebuild in text mode, but it's not necessary. i've done quite a few from x-windows. as for reloading the kernel, it's probably easiest to just reboot. running lilo will also do the job, but i'd highly recommend being in text mode for that. honestly... i don't think i've ever tried a kernel reload from x.
remember, if you get stuck, NOTHING on your system has been changed until you do one of the following:
make modules_install <- this installs any new modules, but doesn't get rid of old ones. it's safe
cp arch.../bzImage /boot/vmlinuz <- this changes the kernel file. but, the new kernel is not yet loaded.
you can run lilo just by typing lilo. i'd recommed rebooting tho - it's just less stressful for newbies.
if you boot from a floppy, then you'll need to change the respective files on the floppy - it's pretty much like the /boot directory.