1) No, an Ultra ATA drive will not work - it must be SATA.
2) Before you change out the internal drive you want to make sure your backup works. I use SuperDuper, so am only assuming CarbonCopyCloner creates a bootable back up also. Re-boot with the back-up attached holding down the option key. Select the external drive, boot up from it and make sure everything is working OK. "Then" you're good to change out the internal.
3) Once you have the new drive installed - you'll do the same - boot from the external drive, open CarbonCopyCloner and clone that drive to your internal.
Not sure about CarbonCopy, but with SuperDuper, there is no need to even open Disk Utitlity as SuperDuper erases and formats the drive when it creates the clone.
Also, even without a backup drive, if you have an enclosure to put your now old external into (meaning the enclosure accepts SATA drives as most enclosures for sell out there are still IDE / ATA only) - you do the same - install the new drive, put your old one into an enclosure, boot from the external and then clone it to the new drive.