How do I make sure the data on my PS3 is backed up on the external hard drive. Furthermore, does the hard drive have to have a certain amount of space available on the drive in order for it to take the backup?
So first off the drive you are backing up to, in good practice anyway, should have the free space of which is used on your PS3. If you used 200 GB on your PS3 try to make sure you have 200 GB free on your back up disk. This backup disk also needs to be formatted to FAT-32 for this to work, this can be done very simply on a computer, plug it in, right-click and format and you will get the option to put it in FAT-32. After the backup is run on the disk you can plug it into your computer and browse through the files on it, I upgraded my PS3 drive before and like you wanted to be sure my files were there before I transferred it. You won't be able to do much of anything with the files on your computer though just as a warning.
I'll admit that I haven't used the backup utility for some time but you should have enough space on the hdd to mirror your current drive. In most cases 500 GB is enough(unless you upgraded to a terabyte or something) and the USB drive must be formatted in Fat 32. The backup usually shows up in the PS3 directory as files on a normal PC.
The backup itself can only be restored to the exact same PS3 that it came from so if your machine dies it becomes useless...