It's not normal behavior. Since you apparently don't want to post a screenshot showing the Sidebar, Path Bar, and Status Bar information and I'm tired of asking questions about it, I suggest starting a new Time Machine backup. Put the existing backup device aside. Determine if the new backups perform as you expect they should. Maintaining two or more backups "in rotation" is advantageous anyway, since one of them could fail at any time, without warning.
The files and folders shown in your Macintosh HD window - such as Applications (Mac OS 9) indicate the information on that Mac was upgraded from much older ones, probably incrementally, over a period of at least ten years. There have been several iterations of Time Machine starting with OS X "Leopard" and each one incorporated minor changes.
Before starting a new Time Machine backup, I suggest you move the folders that ought not to belong at the root level (everything except Applications, Developer, Library, System, and Users) to your Home directory. OS X doesn't rigidly force the user to put specific items at specific locations - for example there is no reason you can't put documents in your Movies folder, or your iTunes Music library in Applications, but to do that would be unconventional. It's also unconventional to put files or folders that ought to be in one's Home folder at the top level of your hard disk hierarchy, so Time Machine may behave unexpectedly.
my opened Current Jobs folder is gone / renamed to Admin.
I don't think that's exactly what is happening. From what I can tell, you are "entering Time Machine" with the CURRENT JOBS folder selected, and it's opening in your Home directory, named "Admin". Nothing is being renamed, but unless you change the Finder's View menu to Show Sidebar, Show Path Bar, and Show Status Bar, you won't be able to navigate to it. That's the reason for relocating it and the other folders as I described.