Hover your mouse over the left end of the google search box and see if the cursor changes to a slider arrow. The search box is width adjustable, maybe someone just adjusted it all the way to the left to take up the entire width.
What you are showing is the Google Search function.
Go to ‘tools’ add-ons and remove ‘Google search’.
I didn’t notice the picture you posted.
Someone simply removed the address block from the address toolbar.
Each piece of a toolbar in FF is modular, you lost that module.
right click on any empty space in that toolbar, click “Customize...” then select the “restore default set” then click “done”.
You should be good to go.
When I put my mouse cursor in between the Google box and the address box, I see a double-ended arrow for resizing the Google box.
If you put your mouse cursor in the small space to the left of the Google box, do you see an arrow? If so, drag to the right.
Nah, it couldn’t be as simple as that.
Switch to MS Internet Explorer.
Dump the Google toolbar.