During the bridge scene where Tucker is shooting the crew members under the Orion women's influence, he is wearing an Enterprise mission patch on his flight suit. In the rest of the episode before and after this scene, he is wearing the Columbia mission patch.
After Tucker shuts down the engines Enterprise is seen to come to slow down and lose attitude control while in orbit. As attitude is controlled by thrusters and no force is needed to maintain orbit the loss of the warp engine would have no effect on the ships orbit or attitude. Even if the ship blew up the pieces would continue in the same general orbit the ship was in.
When they are in sick bay attending Commander Kelby, Phlox injects him with a sedative; a few minutes later when Phlox admits he is also affected, he injects himself, saying he needs stimulants to keep going. He injects himself with the same hypo he injected Kelby with, which has a sedative in it.
During a bridge scene near the end of the episode, phaser shots are fired. In one shot, half of the set wall behind Archer and an Orion woman is missing; rigging and a ladder can be seen in its place.
If the objective of the orions was to deliver Archer, alive or dead, to the Orion Sindacate, they could have killed him and run away the first time they hosted him on their spaceship, at the beginning of the episode.
When Trip confronts Commander Kelby in Engineering about the Orion woman he is showing around, he mistakenly calls him "lieutenant."