Time Machine Backups gone since Yosemite install?

Question

And on a second, possibly related, issue (as discussed in this thread: TimeMachine backups are not accessible after the clean install of Yosemite):
Just had a weird glitch with Dropbox recently whereby files and folders disappeared but I thought "It's ok. Everything is Time Machined. I'm fine!" only to discover that my backups on the Time Machine only extend to the last week... which is shocking, to say the least.
I know for a fact that a few days ago I saw that my 750GB external TM drive was down to about 8GB free and now there is suddenly 82GB free, plus a never before seen folder called "lost+found" next to the Backups.backupdb folder. What is it and where did it come from?
Just noticed now that my 750 GB drive holds 13.68 TB of data (that is seriously impressive), and that the lost+found folder is 22.66 GB big.
Fortunately, Dropbox has the "Restore Files and Folders" feature so hopefully nothing is lost but this Time Machine glitch is a horrible eye opener - all of a sudden I cannot rely on my backups extending back at least a few months, like I have entrusted it to do for years.
Since upgrading to Yosemite I have had a whole bunch of... seemingly trivial... glitches and bugs (dropbox syncing errors, re-establishing Apache and a local test-server issues plus a few other minor irritants plus the unexpected random reboot of the computer twice in the last 2 weeks) that have now all added up to leave me anxious about what new problem is going to manifest.
For example, in trying to view any Time Machine logs that might give a clue as to when or why the backups have disappeared I opened Console only to find that the console window doesn't appear. Anywhere. Even new ones.
I'm not a geek and have no idea how Console might help me but i was simply following the instructions in this link http://www.macworld.com/article/1131534/tmeye.html but Yosemite is not doing what I think it should.
This is what that app LogViewer Time Machine displays (seriously, my Console doesn't show a window - how do I fix that?) about the earliest backup, which is from Tuesday, Nov 18, 2014.
8:28:28 am - Starting automatic backup
8:28:35 am - Backing up to /dev/disk3s2: /Volumes/BackUp Brain/Backups.backupdb
8:28:43 am - Will copy (1.53 GB) from Macintosh HD
8:28:43 am - Found 1321 files (1.57 GB) needing backup
8:28:43 am - 3.72 GB required (including padding), 15.03 GB available
8:35:24 am - Copied 1530 items (1.57 GB) from volume Macintosh HD. Linked 7172.
8:35:43 am - Created new backup: 2014-11-18-083543
8:35:45 am - Starting post-backup thinning
8:35:45 am - No post-backup thinning needed: no expired backups exist
8:35:45 am - Backup completed successfully.
This one from this morning throws up a weird error - seems like my iMac has developed a number 2 after it's name (I will avoid the obvious joke that if these glitches keep occuring under Yosemite then it's very apt):
8:16:19 am - Starting automatic backup
8:16:19 am - Backing up to /dev/disk3s2: /Volumes/BackUp Brain/Backups.backupdb
8:16:37 am - Will copy (2.5 GB) from Macintosh HD
8:16:37 am - Found 2730 files (2.55 GB) needing backup
8:16:37 am - 5.44 GB required (including padding), 38.76 GB available
8:24:45 am - Copied 3029 items (1.97 GB) from volume Macintosh HD. Linked 11142.
8:24:52 am - Created new backup: 2014-11-24-082451
8:26:05 am - Starting post-backup thinning
8:26:05 am - Error: Error Domain=NSOSStatusErrorDomain Code=-50 "The operation couldn’t be completed. (OSStatus error -50.)" (paramErr: error in user parameter list) deleting backup: /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-24-081624.inProgress/B129B56F-A2A5-46DB-B181-719D23BF3FE5
8:28:15 am - Deleted /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-23-055154 (1.41 GB)
8:28:36 am - Deleted /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-23-043709 (1.1 MB)
8:28:46 am - Deleted /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-23-033533 (868 KB)
8:29:32 am - Deleted /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-23-023006 (1.9 MB)
8:30:28 am - Deleted /Volumes/BackUp Brain/Backups.backupdb/Diarmaid’s  27" iMac (2)/2014-11-23-012426 (1.1 MB)
8:30:28 am - Post-backup thinning complete: 5 expired backups removed
8:30:29 am - Backup completed successfully.
Any help or assistance or guidance would be deeply appreciated. Haven't had any major issues with this machine since I got it, till the Yosemite update.

Answer 1

Just back from an Xmas break in the wilderness and took a look at my Mac. Firstthing I see is that Time Machine has not backed up since 24th Dec. LogViewer for TimeMachine reports the following:
12:15:13 am - Starting automatic backup
12:15:14 am - Error determining directory contents for /Volumes/BackUp Brain: Error Domain=NSCocoaErrorDomain Code=256 "The file “BackUp Brain” couldn’t be opened." UserInfo=0x7fa539607ad0 {NSFilePath=/Volumes/BackUp Brain, NSUserStringVariant=(
    Folder
), NSUnderlyingError=0x7fa539607e30 "The operation couldn’t be completed. (OSStatus error -36.)"}
12:15:14 am - Backing up to /dev/disk3s2: /Volumes/BackUp Brain/Backups.backupdb
12:15:15 am - Error -36 applying backup protections to /Volumes/BackUp Brain/Backups.backupdb/Diarmaid‚Äôs  27" iMac
12:15:15 am - Error: (22) setxattr for key:com.apple.backupd.HostUUID path:/Volumes/BackUp Brain/Backups.backupdb/Diarmaid‚Äôs  27" iMac size:37
12:15:15 am - Backup failed with error 2: 2
With variations of the same till today. My 750GB WD Elements external is still there visible on the desktop, displaying 66GB free with the "lost+found" folder and the Backups.backup folder, only this time neither is displaying any contents.
Seriously, does anyone have any idea what is going on with Time Machine? If I cannot rely on it for automatic backups I don't know what I am going to do. It has ALWAYS been there, working perfectly in the background and saving my *** numerous times.