Why do you assume Parker testing his webshooters would be cut? It would fit in the movie just as well as the scene where Tobey Maguire shoots organic webbing all over his room.
What about Dr. Ock? His powers werent biological, they were mechanical. The movies allowed for mechanical marvels such as Green Goblin's glider and Dr.Ock's mechanical tentacles, I dont see what so absurd about a webshooting device.
Just like Sam Raimi, you underestimate Peter Parker's skill and resourcefullness. Parker is smarter than you and me. He routinely matches wits with mad scientist and evil geniuses. He can whip up a special web formula to counter some villians special power in minutes. If you made a movie about Sherlock Holmes would you dumb down the character to make him more believable? No, that would defeat the purpose of telling the story.
Parker created the webshooters and webbing formula in days. That what the character is capable of. He's not your average teenager as the movies would have you believe.
Peter Parker makes a million dollar costume without any knowledge of tailorship and the audience believes it right away. Just like time travel in the "Back to the Future" films people accept science-fiction because its fun and imaginative. A time traveling car is silly, but the fact that its a cool idea causes ppl to go along with it. The truth is the movie universe allowed the quick invention of webshooters by a teenager. It doesnt require some elaborate fantasy world. The idea just simply needed to be introduced and the audience would happily go along for the ride.