Please accept my apologies for making this post so long, but it is the best I can do in offering a solution to your issues.
I don't use Time Machine at all, and on my MacBook Air, never seen anything like .MobileBackups. Again, I use SuperDuper! for my backups.
Also, that display you show via "About This Mac" is definitely strange, and the real weird one is the 389.69 GB of space designated as "Apps". Myself, my "About This Mac" display is definitely more straight forward. Here is what it looks like:
Note that I don't have Other "broken out" like you do, ie, Apps, Audio, Movies and Photos. I don't even know how to do that!
If you want to "clean things" up, here is one possible approach:
1. Download the "free" version of SuperDuper!. You can get it from here:
http://www.macupdate.com/app/mac/13803/superduper
Don't worry about the price (unless you want to purchase it. I eventually did).
2. Install SuperDuper! onto your internal hard drive.
3. Use Disk Utility to add another partition to your external hard drive. I would make the size of it about 30 to 50% larger than the primary partition on your internal hard drive. From the picture you provided, that seems the only "visible" partition you have, and it looks like you are using 138 gig of space. So, make the size of that partition about 180 to 200 gig. I would name it "Macintosh HD Backup", but that is up to you.
4. Download the excellent freeware program called Recovery Partition Creator. You can get it from the "top" link on this link:
http://musings.silvertooth.us/2014/07/recovery-partition-creator-3-8/
Don't worry that it says compatible with Mavericks. It works fine with Yosemite. You'll see later the reason why you need this.
5. Make sure that you have the file called "Install OS X Yosemite" both inside your Applications folder, and somewhere else on your internal hard drive. If you don't have it, re-download it from the Apple Store (that download automatically places that file in your Applications folder) and then make a copy of it somewhere else on your internal hard drive.
6. Launch SuperDuper!, and make the backup of your stuff on the internal hard drive to the Macintosh HD Backup (or whatever you called it) partition on your external drive. Depending on the type of connection you have, this could take some time. For me, when I was using the free "version", since that external 1 TB drive is connected to my Mac Mini via Firewire 800, it took about 35 to 40 minutes to backup that 80 gig of "stuff". But, for my other hard drive that is connected via USB 2.0, it took close to an hour (those are similar times for my MacBook Air). Since you are backing up more (138 gig), I suspect it will take longer.
7. Either after SuperDuper! completes, or after restarting your Mac, go to Systems Preferences, select "Startup Disk". Then, select the icon for the "Macintosh HD Backup" (or again, whatever you called it), click Restart, and your Mac will start up from that external partition.
8. After booting from that external partition, run Disk Utility form there to 1) Erase your internal hard drive (I would choose a "speed" somewhere in the middle of the scale), format the just erased drive, and if necessary, partition it. Call your primary partition (maybe only one) the same name as you originally used, "Macintosh HD".
9. Quit Disk Utility, then launch SuperDuper! and do a restore of everything from that external back up partition you used for SuperDuper! to that partition on your internal drive.
10. After that restore is complete, go to Systems Preferences, select "Startup Disk". Then, select the icon for the "Macintosh HD" (or again, whatever you called it), click Restart, and your Mac will start up from your internal drive.
11. Hopefully, all that works! (I actually had to do that twice on my Mac Mini, and it worked like a charm!
11. Now, at this time, there is an invisible partition on your internal disk called "Recovery HD" (you can make it visible in Disk Utility). As the name implies, it can be used to recover from a disaster (it has other uses also). When SuperDuper! makes the backup, it does not back up that partition (only Carbon Copy Cloner (another cloning program) does, but that one does not have a "fee mode"). When Disk Utility erased the internal drive in step 8 above, it erased everything, including that Recovery HD partition. There are a couple of ways of re-creating it, but using that program "Recovery Partition Creator" you downloaded in step 4 above is the easiest. Make sure to follow the directions, and make sure you point to the "Install OS X Yosemite" file inside your Applications folder. That will create the Recovery HD partition. I have used it successfully twice (with Yosemite).
I would think that after doing all that, and turning off Time Machine, your initial issues will disappear. Going forward, use SuperDuper! to make backups as often as you wish. (You will actually replace the last backup made by SuperDuper! with a new one). I use it once a week, but if you need to run it more often, go for it. (The paid version does have a "scheduling" feature so that SuperDuper! will behave like Time Machine with the backups, but I know hardly anything about it, as I don't need it). At some future point, whenever you feel comfortable, and as long as there are no "old" files that you need, you can delete all the Time Machine backups that have been made, and thus free up some space on your external hard drive.
Finally, as I mentioned, I do 3 other things first before running SuperDuper! as part of my disk maintenance/clean up/backup processing. Here is what I do:
1. Get rid of the deleted EMails that have been accumulating via my EMail program, Outlook 2011.
2. Run the excellent freeware program Onyx to clean out some unneeded "crap", along with doing a few other maintenance tasks. I have been using this program for eons, and I cannot live without it! In fact, whenever there is a major Mac OS upgrade (like going from Mavericks to Yosemite), I wait to make sure that either a new version of Onyx is released, or confirmation that the latest version (I make sure to have the latest version; in fact, I do that with all the software I have) does work with the new OS. Then, I do the OS upgrade.
3. It is a good idea to have at least one other Disk Maintenance/Repair program that is more robust than Disk Utility. I actually have both Disk Warrior and TechTool Pro, and I use TechTool Pro to do some more Disk Maintenance/Cleanup tasks. (I run Disk Warrior once in a while). One can do some of that with Disk Utility, but most likely, Disk Utility would need to be launched externally (Disk Warrior would also need to be launched externally). But, the neat thing about TechTool Pro is that it creates a bootable, small partition on one's internal disk, called an eDrive, and via TechTool Pro, it can be launched to do all the disk maintenance/cleanup tasks as if it was doing it externally. Just like Onyx, I have been using this program for eons, and I cannot live without it!
All of that might seem like quite a bit of work, but it is worth it, as I hardly even have issues with my machines.
One other comment is needed. There are some folks that swear by Time Machine for doing their backups. I do understand where they are coming from, especially for the need to retrieve a specific file (or files) from a recent backup. For those folks, having all those incremental backups is required. But, then there are folks like me that don't need such incremental backups, and thus using a cloning program like SuperDuper! or Carbon Copy Cloner is easily enough. There is one distinct advantage to using a cloning program and that it is a bootable backup is created, so it is quite easy to recover from a disaster. I have no idea how (and if) that can be done from Time Machine backups.
