Input/output errors at start-up
 

I was told that this error could indicate some bad blocks. Since I can't safely run e2fsck on a mounted file system, how can I?

Do you have a boot/rescue cd? You can use it to boot and then run e2fsck. It's usually the safest way. Also, you can unmount a filesystem (if it's not your / or another important one), check it, then mount again.

input/output errors
 

I tried sudo umount /boot; and got device is busy(The same error for /home)
From Your suggestion (I could use Sarge (Disk #1) or Knoppix?
With Knoppix, when it first starts, everything is unmounted. Would I then open an Xterm, and do e2fsck -c?

That's about it. Another way, which I usually use, is to boot into signle user mode, and do "mount -o remount,ro /" to remount the root partition readonly (unmounting everything else). You can then run fsck and badblocks safely.