If you right click at the top area in firefox then select customize you can drag the little search bar you never use to the customize window and its then gone.
it will make the main url bar much longer to fill the space ,
from the customize window you can drag the space icon to the right of the big url bar to shorten it, you will have to drag the space several several times untill you have it how short you like. heres a screen of my top bar minus annoying little search bar. also if you drag the google homepage to your home icon/button then thats your homepage and there is an add on for ff to make a new tab open with your homepage.