When Watson and Holmes enter the offices of Shad Sanderson, the New York time shown is 07:45. A few minutes later, when Holmes is darting around the office after his meeting with Sebastian, the New York time shown then is 07:21.
Holmes is "backstage" while the Chinese circus is performing. He sprays a single, yellow line across the mirror. He hears something behind him, spins around, the camera angle changes to include what Sherlock hears, with the mirror now behind him. On the mirror is now a doubled spray of the line, like he went back and forth over it, instead of the single line.
When John and Sherlock arrive at the National Art Gallery, there is a crowd of people at the bottom of the steps. When Sherlock says that he needs advice, the crowd disappears and reappears between shots.
When Sherlock enters the lounge with Detective Inspector Dimmock the forensic analyst who is passed the plastic bag has a mask on his neck, it disappears after he walks out of the shot.
The website "Online News" is actually a local file on Sherlock's laptop, not a website.
About 22 minutes into the episode, when Sherlock is showing Watson the Online News website concerning the mysterious killing, the next headline reads "Euro referendum rears it's head" (also on the summary on the right), an obvious grammar mistake for "rears its head"
The specific type of handgun (here: semi-automatic) has next to nothing to do with the muzzle velocity of the projectile.
Sherlock deduces that a character has crossed the international date line twice in the preceding month, because the calendar feature on the person's watch is two days behind. However, watches know nothing about the international date line -- they count/measure elapsed time according to their spring-driven mechanism or crystal-frequency electronics. The calendar error created by crossing the international date line (first observed by Magellan's crew) is a human one, caused by reckoning days by the number of sunrises observed, rather than the number of ticks of a mechanical or electronic watch.
If anything, a high muzzle velocity makes it less likely for a projectile to ricochet off a wall since it has more energy and thus is more likely to penetrate or dig into a given surface.
When Sherlock is searching for "Chinese antiquities sold at auction", his address bar shows a local address, not an Internet one, so he's actually just watching files on his laptop.
When Shan is taunting Watson with her pistol, she loads it in front of him, and racks the slide to load a round into the chamber (albeit, she didn't rack it far enough back to actually do so). A few moments later, when Holmes makes his appearance, Shan is seen racking the slide again. However, no round is ejected (the one she loaded moments earlier) revealing the pistol to be unloaded.
The "Walther P5" used by the General is actually an ME8 blank firing pistol made in Italy.
When Holmes is darting about between desks in the bank, he bumps into a substantial column with a currency exchange board on it, causing both to wobble conspicuously.
When Sherlock enters the Banker's flat and spots the body there is a flash of a gun (Sig Sauer P226R) on the floor, if you look closely you can see this gun is missing the top grip screw which coincidentally was also missing from the Sig Sauer P226R in Watson's drawer in the previous episode indicating the same prop was used for both scenes.
The handgun used by General Shan in the tunnel scene is an Italian Bruni "New Police" top-firing blank pistol. It is in fact a starting pistol, and is quite incapable of firing bullets as it has no barrel.
The graffiti artist is apparently painting in the same style made famous by Banksy. This style gets its distinctive look from the fact that it is made by using a stencil. This is so the artist can quickly apply their work and move on without being spotted. Not only did the graffiti artist not have a stencil, but they apparently were taking so long that a friend would know where to find them.
Several times Sherlock refers to the ancient (fictional) Chinese writing style as a dialect, but Chinese dialects are restricted to how the language is spoken, not written. It would have been more correct to indicate that was an obsolete version of the characters.
Sherlock calls the ancient numbering system "Hang Zhou" pronouncing it as "Hang Zoo". The correct pronunciation would be closer to "Hahng Joe" which is something Sherlock should know.
Sherlock states that, "Exit visas are scarce in China". This is untrue. Chinese people are granted an exit visa the same as most countries through the possession of a passport which is a simple procedure for most Chinese citizens.
Soo Lin generally speaks with a typical Chinese accent throughout most of the episode yet in the beginning, she says, "Is that security?" in a decidedly English accent.
Watson's behavior when investigating the source of the gunfire seems inconsistent with his experienced military background and PTSD, such as slowly walking through the middle of open hallways without maintaining cover.