Designated Hero: As any history major can tell you, many of the the rulers you're working for aren't looked back upon too fondly by today's historians, but when you're their loyal city administrator they are universally virtuous and wise. Also, as far as the mission briefings are concerned, anyone who isn't a part of China and doesn't serve the Emperor loyally is a barbarian and should not be trusted.
Disappointing Last Level: The final level of the game is pretty much more of the same — build a monument, have 72 months of heroes, a population of 6,000 people, and 250 people living in Heavenly Compounds. The only difference is the very aggressive and powerful Mongolian Empire that's likely to launch an invasion or two, but you can make them an ally to take them out of the equation.
Game-Breaker: Many Heroes come with benefits that snap the difficulty of a mission in two.
Sun Tzu (appropriately) makes military actions much easier. He halves the construction costs of forts and fortifications, makes it cheaper to hire spies, captures any disguised enemy spy he finds patrolling the city, reduces the travel time of armies, improves infantry morale, can bless a fort to double the training speed of troops, and when in combat himself he's one of the best offensive heroes. If you plan to attack an enemy, Sun Tzu can make or break the attempt.
Guan Di can bless a fort with free loads of weapons or warehouses with free loads of bean curd, while Xi Wang Mu can bless a jade carver's studio with a stock of raw jade. Build a row of warehouses/forts/carver studios and send them down the road handing out blesses for tons of free goods.
Wild animals that attack walkers, namely salamanders, gobi bears, vultures, tigers, and alligators. They hunt down and kill walkers, and if they happened to be carrying any goods, those goods are gone. One mission has the spawn point for visitors to the map a few tiles away from where salamanders hang out; have fun watching immigrants and traders get eaten trying to pass them to get to your city. On many maps you can just build a couple of sentry towers near their respawn point and let the sentries keep them contained. Can't build sentry towers? Gonna have to arm some troops and march them out there.
Spies. Wondering why your stocks of weapons/ceremics/hemp seems to go faster than you use them? Stick a guardhouse by your warehouses; there's a spy running around stealing your goods.
The game handles attacking rivals oddly if you conquered them while their army was on the way. Bribing the rival army suddenly makes the vassal a normal trade party, and fighting them off keeps things normal and surrendering makes you their vassal as expected. When it comes to a city that has conquered you attacking to quell your rebellion, if you fight them off it's possible their city will become your vassal as a result.
Feng shui doesn't change on a building once it's been placed, so you can build a structure near a group of trees so it has perfect feng shui, then destroy those trees to change the feng shui of the surrounding tiles to something more buildings need, while the original building remains in balance.
Narm: The narrators of the various mission briefings pause when stating the numerous city and character names. Coupled with the sudden changes in tone that come when they do this, it sounds like the voice actors either stumble over enunciating the names properly, or that some Mad Libs-esque editing was going on with the pronunciation of the names inserted after the initial recordings.
The Ancestor heroes. The other three pantheons have their heroes always at Contented at the least, but Ancestors will fall to Neglected, Unhappy and eventually Angry, and when they get Angry, they may decide to unleash a natural disaster on you as punishment for not paying them lip service. Keeping them happy requires regularly sacrificing goods to them (and they like expensive stuff), and bad feng shui makes sacrifices less efficient.
You can set city gates to allow or disallow market employees, inspectors, or religious walkers to pass, but you cannot set any restrictions for other walkers, namely guards, tax collectors, herbalists, water carriers, and acupuncturists. Those walkers are always not allowed to pass, creating a lot of headaches when trying to build your city and a misplaced gate cuts your city off from a critical walker.
Messages sent to other cities at the same time don't arrive in the same order you sent them. For example, say you send an ally gifts of goods to improve your standing with them, and after that send a messenger asking for a trade agreement or alliance treaty. More often than not they'll refuse the latter first, then receive your gifts and your standing improves to the point they would have agreed if they had responded to the trade/alliance request afterward. Now you have to send the request all over again and wait several more months for a reply.
Even standard housing requires multiple food types, which is often inconvenient or expensive to maintain. While it's possible to set a market's minimum food level, it means they won't go to any mill that doesn't have that level of food, meaning it's possible for housing to regress to the second-lowest level with full granaries and idle markets.
That One Level: A lot of the levels set in the desert. They usually have one large grassy area, limiting where you can build your city (Wells are needed to get beyond the most basic housing levels and can only be built on fertile ground), forcing you to build several more farms than usual due to the infertile ground limiting their growth, your only reliable source of income is probably going to be salt and whatever metal of the age your smelters produce, and occasionally ceramics. Furthermore the earlier missions which use Wood for tax collection require you to import it since trees of course are of short supply in a desert. A Han mission even has you building the Great Wall, which requires hundreds of units of Wood, thus meaning you're going to spend the entire mission importing it and have to come up with exports fast so you can afford to.
Tier-Induced Scrappy: Crossbowmen are nearly useless. They come in groups of 16 and (obviously) are armed with crossbows, letting them engage enemies at range and set buildings on fire, but they're terrible in melee combat. However, calvary are also armed with crossbows for the same combat advantages, move much faster than crossbowmen, and are better in melee combat. The advantage crossbowmen have over calvary is their crossbows do more damage and fire further, but the calvary's superiority in all other areas outweighs those stats, and the calvary's crossbow fires faster anyway.
They Wasted a Perfectly Good Plot: The conquest of the rest of China by Qin Shi Huang is entirely glossed over; save for the first mission where you build the great canal a few decades earlier, the Qin campaign focuses on you building up the empire and building some monuments in the capital, when it could have included at least one mission that gave attention to his conquest of the other enemy states.