New OEM (F6PZ9H307DB) part is installed and the fuel gauge works perfectly now. The float is made out of some composite material, not an empty barrel anymore, although I believe the sending unit (electronic part mounted on the pump assembly) was defective. I now have a reserve pump, just in case I get stranded anywhere and I should be able to swap it myself in a couple hours. The ratchet straps worked like a charm to lift the gas tank back in place - no jack or forklift needed.
Now I got another problem - the tires were half-way deflated. When I pick up speed at about 20 mph I get a drop-like vibration where the front end (especially under driver seat) shakes as if the tire was not perfectly round. One mechanic who inflated my tires said it could be caused by a "flat-spot in the tire due to sitting for 3-4 years, and being exposed to sun). Tomorrow I will replace the tire with the spare wheel and see if it goes away. Any idea what kind of jack I should use to swap the tires? Anyone ever had similar shaking of the front axle?