I didn't think I'd ever need to say this, but - yes - pilots are indeed required to verify that an aircraft is fit for flight (ie airworthy), before making a take off.
Pilots are required to complete an inspection, which is one part of a larger system of inspections, maintenance, manufacture, and design.
A successful preflight inspection by no means ensures an airplane is airworthy or safe. It takes a lot more money, paperwork, and time to make sure an airplane is airworthy.
And there are numerous DC-3's still in operation worldwide, some of them dating back to the 1930s.
Yes, and many/most of them are operating under the intended, appropriate category of airworthiness certificates, with continuing oversight including ADs, etc. You mention "regulatory oversight" of this bird, while conveniently omitting that it is operating with an Experimental category certificate...which means it gets vastly less oversight...and that said certificate requires it not be flown over populated areas, which they've clearly violated at least in spirit.
Operating under an Experimental certificate means that Red Bull is taking all responsibility for the airworthiness of an airplane the manufacturer/FAA/government operator decided was no longer economically viable. This could mean that Red Bull is spending serious time and money ensuring the continued safety of the bird, or it could mean that there is a lot of whistling past the graveyard going on. I don't personally know about the program or even care: I just know your comments about airworthiness don't apply to pilots after Wilbur or Orville.
SRSLY, B-52s? You are comparing the operation of aircraft with the full force of the US national defense to a single aircraft operated without the manufacturer's oversight by a soft drink company?