Everything Apple (the Mac heads thread)

You've got at least three independent problems here - the DVD is not working, you aren't able to log in to iCloud and the external drive won't mount. Treat them separately but the combination of the two latter issues leaves you vulnerable to major data failure.

Start with getting access to iCloud. Did you log out of iCloud and log back in? You shouldn't lose any local data doing this. Can you now see your iCloud? If not, could you create yourself a dropbox account and back up your files there temporarily?

Then you'll need to start with some troubleshooting for the external drive:

1. check if the external HDD is functional. Plug it in to another mac. Can it be seen in disk utility? If not, change the caddy to see if that is the issue.
   
2. The USB ports on the iMac. Are any of them working? Try some other devices. if so, get yourself a big USB pendrive and back up your files locally.
   
   

If you can't mount the drive you won't see it in Time Machine.
More importantly I wouldn't rely on Time Machine for resilient backup. It's proprietary and somewhat flakey. Effectively it has been replaced as a backup service by iCloud.

If you can mount the drive (and better, also a new external drive or pendrive) you can use time machine or carbon copy cloner to clone the original drive and ensure that whatever happens you can fully recover your machine. If you can't mount anything over USB your iMac is probably fuxxed and you must get iCloud working to back up your files before the SSD fails. There's a chance that a full OS rebuilt will fix it but obviously you can't attempt that until the data is backed up.

One thing that is often overlooked is that SSDs don't fail gracefully like mechanical disks. When SSD's go, they just die and there is almost no chance of recovery. Mechanical disks typically degrade such that individual files get corrupted, but the rest are often recoverable.

If this were my machine I'd backup to iCloud and, if possible, make a separate Time Machine backup AND copy critical files to a pendrive, then wipe the internal and rebuild the OS. This might fix any ongoing USB problems you currently have. If backup to iCloud or over USB was not possible working I'd buy a replacement mac and copy the files over from iCloud, or from the iMac using target disk mode over thunderbolt.

You can do all this yourself -it's time consuming but not complicated - just be methodical and don't rush it. A checklist to work off as you go is really helpful.