After the gun battle in the hotel, the thugs shot numerous holes in the walls. You could tell by the way the fragments around the holes were showing that the bullets didn't enter the walls but exited the wall by charges planted inside the walls.
At 10:03 we see Joe at the docks walking on the alleys and passing by a guy carrying a crate on his shoulder, than by another guy carrying a canister, while turning right. The same scene appears in episode 02 at 33:48.
The MCRN frigate Yangtse dispatches an inspection skiff to the Rocinante and orders them to submit to inspection, even though the status display shows the Rocinante is still well on the United Earth side of the border. A Martian Navy warship would not have any authority or jurisdiction over a ship still inside Earth territory.