Corrected entry: It is established at the start of the movie that the Klingon Bird of Prey is the same one that they flew to Vulcan in Star Trek III: Search for Spock. The bridge of the ship however is totally and utterly different to the one shown in the previous movie. There can be no claim to suggest they remodelled it in their time on Vulcan, because everyone still uses the controls with a degree of caution, which they wouldn't do if they modified it to suit their needs.
Corrected entry: When the crew starts to evacuate the Bird of Prey from the ship's hatch, the sound stage wall is easily visible.
Correction: That's the wall of the escape hatchway, not the sound stage wall.
Corrected entry: Starfleet has serious regulations on bringing future technology to inappropriate time periods (which is why the Bird of Prey remained cloaked during it's time in 1986). Chekov being fully aware and considerate of these regulations, considering he has been in Starfleet for about 20 years should have never tossed the Klingon phaser (23rd Century technology) at the navy officer (20th Century person) with full knowledge that he'll never be able to retrieve it. He is too smart for it to have been panic.
Correction: Maybe a bad error in judgment, but a character mistake, not a movie mistake.
Correction: First off, the men interrogating him do not appear to be Navy officers but rather federal agents (FBI, CIA, etc.) The guy asking the questions has some idea of what the phaser is supposed to be, "Make nice, give us the ray gun" so he either deduced that or as is common in interrogations, they're asking the same questions over and over. The interrogators obviously don't believe a word of what Chekov is saying, hence why the one agent tells him, "Go ahead, stun me."
Correction: There most certainly can be made a claim that they remodeled it on Vulcan. What difference does it make if they, "use the controls with a great deal of caution?" People use brand new things and things they are already familiar with a great deal of caution. And this measurement is highly subjective at best. The crew is likely a bit apprehensive because the "guts" of the ship, for lack of a better term, are still Klingon. Scotty even speaks negatively of the dilithium crystals in the ship at point.