My external hard drive won't do anything...

So, here's the deal:

1) I updated OS X this morning.

2) I went to backup to my external hard drive (LaCie rugged) through time machine.

3) It appeared on my desktop, as normal. I started the backup and it keeps telling me the disk ejected and stops the backup.

4) I have restarted my computer multiple times, and no luck.

5) It now will not appear on my desktop.

[Enter google]

7) Learned about Disk Utility.

8) It appears in Disk Utility (problem solved? YAY! just kidding, no it's still not working)

9) I go to "verify", nothing happens

10) I try to "mount", no luck.

11) I try to erase, it tells me there is an error and to back up my files (which I can't do, I back them up to the hard drive that's having issues)

12) I restart in Recovery mode, still doesn't work.

[Enter google, again]

13) I attempt to use terminal to solve the issue. I have no idea what I'm doing, but it doesn't change anything.

14) I unplug, replug in, Nintendo 64 blow on the ports to make sure there is no dust, NOTHING works.

**Below are screen shots of disk utility. I've tried everything I know how to do (and some I had no idea what I was doing) through the options listed in the photos. HELP!!!

15) WHAT DO I DO?! I'm about to throw my laptop out the window.