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Follow-up:

The problem while trying to start a Backup you see below. This doesnt make sense, as the external drive has 999gb...

1. it gave itself a new cryptical name, and I can't change it back
2. in most cases I cannot install new software anymore (e.g. I cannot update Flash)
3. It is sometimes shutting down immediately
4. Disk Utility says "no errors found" but can't mount it.

diskutil cs output from Terminal:

output from Terminal for command sudo gpt -r show /dev/disk0

output from terminal for sudo gpt -r show /dev/disk1

my counteractions so far

1. ran the Apple Hardware Test. Result: nothing found.
2. ran the First Aid (see screenshot). Result: nothing found.

Any help is much appreciated! Wanna get rid of this :(

1. It gave itself a new cryptical name, and I can't change it back
2. In most cases I cannot install new software anymore (e.g. I cannot update Flash)
3. It is sometimes shutting down immediately
4. Disk Utility says "no errors found" but can't mount it.

diskutil cs list output from Terminal:

Output from Terminal for command sudo gpt -r show /dev/disk0

Output from Terminal for command sudo gpt -r show /dev/disk1

My counteractions so far

1. Ran the Apple Hardware Test. Result: nothing found.
2. Ran the First Aid (see screenshot). Result: nothing found.

No backup available!

Any help is much appreciated!