I recently found that a pipe looking piece on the underside has rusted out so badly, its about to fall off. My question though is, what is the name of this part (circled in attached image), and exactly how important is it (is it "you need to get it replaced asap important, or have it replaced when you feel like it, or doesn't really need to be replaced at all)
Pretty sure that's a sway bar to make the chassis more rigid when making sharp turns at speed. This is an opportunity to upgrade with a thicker stronger piece of steel, same with that bar in the center of the picture over the catalytic converter. The chassis stiffening kits you can buy online cost a fortune, but anyone with basic metal working skills can build the same thing for 20% of what the retailers charge.