Da Vinci's spacecraft's propulsion system of flapping wings would be useless in the vacuum of space, as there would be no air for the wings to push against.
Da Vinci's spacecraft using flapping wings and a propeller as a method of propulsion is not a mistake, it terms of the theory of its design, in practice it obviously would not function. Most of da Vinci's inventions were created and/or postulated in the early 1500's, at that time humans did not have any knowledge of the environment of space, only theories, and many scientists at the time believed space to be filled with gases (though some also speculated it was a vacuum). So if da Vinci thought space was filled with gas the design of the spacecraft makes sense. By the late 1700's scientists widely agreed that space was almost certainly a vacuum, but it wasn't until the early 1940's when the first unmanned rockets were sent into space, that humanity had incontrovertible proof of space being a vacuum.
The manhole cover (giant coin) goes in Bender who goes in the manhole, an impossible fit by round cover design. Bender's cavity is repeatedly showed to be "magic", but the puzzle designer still seemed to plan for the cover to go through.
The professor said the craft wasn't aerodynamic because it flies in space, yet when it reaches the planet, it flies in and lands.
There's not a day's worth of air in the bubble Fry and the professor are in or a month, it's not even sealed in anyway, not to mention the lack of any other life support.