Strange Time Machine Behavior

I don't have a quick answer for you, but it may help if you were to first show some path information in the Finder window.

To do that click the Finder's View menu, and select Show Sidebar, Show Path Bar, and Show Status Bar - all three.

Then, "Enter Time Machine" and determine if it's easier to navigate.

... whenever I enter time machine it opens in "Admin".

I'm not quite sure what you mean by that. Does that mean TM appears to be opening under another user, named "Admin"?

In that case turn off Time Machine using Time Machine's Preferences. Unmount the backup volume.

Navigate to the following folder by selecting the entire line (triple-click to select it), then control-click, and from the contextual menu that appears, select Services > Reveal in Finder:

/Library/Preferences/com.apple.TimeMachine.plist

Drag the entire folder to the Trash, reconnect your backup device, wait for it to mount, and turn Time Machine on again. This will not create a new backup nor will it erase your existing ones, but the values shown in the "backing up xxx of yyy" message will be inaccurate until TM finishes the subsequent backup.

Determine if that fixes the problem.

Do the Sidebar, Path Bar, and Status Bar appear for you? No need to post a screenshot if you consider the information personal.

Those preferences will remain active in the Time Machine window. When they are present, can you navigate at all to the desired folders in the Time Machine window?

I have to step away for a little while, but this is where my questions are leading:

Time Machine is working normally. The problem you're experiencing is due to the fact that you are attempting to access items at the root level of your hard disk. Updating OS X to version 10.9.3 probably corrected file permissions settings that had been incorrectly set in the past, for reasons unknown and probably impossible to determine now.

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.