When Tucker transfers from Columbia to Enterprise, he heads out of Columbia with his feet first. In the next shot, a close-up, he is climbing "up", i.e. head-first, suggesting that he has already rotated by 180 degrees. Yet in the following long shot, he is still in the act of performing that rotation.
When Columbia approaches Enterprise upside down, a full-screen view of the display showing the approach is shown. In the next shot, the same display can be seen in the background, but both ships are shown to be already next to each other. In the subsequent shot, Columbia is missing completely from the display.
When the rigging fell, it was swept behind the ships as soon as it passed through the launch bay doors. There's no wind to offer resistance and blow it back, so since it was inside the warp bubble, it would continue moving at the same speed as Enterprise. It should have simply fallen down into Columbia's launch bay, just like the grappling hook first did when they shot it across.
At the beginning of this episode, as Trip is just about to transfer from the Columbia to the Enterprise, he is in an EV suit with the Enterprises logo prominently on the front. In the previous episode, Captain Hernandez told him to have the patch on his duty uniform changed. So either the Columbia is borrowing EV suits from Captain Archer, or the prop department goofed.