Your Time machine no longer recognizes your hard drive, however, it can still be restored from.
I suggest that you manually back up those files either to a USB drive or maybe a DVD.
As for the applications, those can be reinstalled.
I know this is all time-consuming and that is an unfortunate thing. If you wish this to not do this to you again or to avoid further problems I do recommend doing it this way.
You have a permissions conflict and few options to correct it.
Your Time Machine disc, should be at a minimum, as your internal hard drive. What you have will probably work but it may also require starting from zero.
The reason for this is because of hardware changes.
Another reason why your Time machine backup failed is because of the permissions conflict.
I wish there were an easy button but, unfortunately, in this case I'm fresh out.