Cast iron was first developed in China in the fifth century BCE. It did not appear in European society for almost two thousand years. Cast iron offered the advantage of being much cheaper and more efficient to produce than steel, although it was also much more brittle.

The earliest accounts of gunpowder and its use comes from accounts of Tang Dynasty China, so that would be the best choice. Europe eventually gained knowledge of gunpowder when it was brought by trade and the Mongol conquests, so the British Empire would be poor choices. Gunpowder was brought to the Americas by Europeans, so the Mayans would not be a good choice. Lastly, the Roman Empire was no longer extant by the time gunpowder was first developed, so they too would be a poor choice.

The Mayans made millennia worth of astronomical predictions and observations that are still well-known for their accuracy, so they would be the best choice. The Phoenicians were a great sailing and trading culture, but they were not known for astronomy. The Akkadians conquered Mesopotamia and were known for their use of iron weapons, not astronomy. And lastly while the Egyptians did have a strong and thriving culture, they were not known for their astronomical predictions.

Separated by thousands of miles and vast oceans, Mesoamerica and Mesopotamia both developed writing systems independent from one another. They were not aware of each other, and certainly didn't cooperate or fight. The evolution of writing was largely a result of agriculture and the need to keep records of surplus.

The compass was invented in China during the Han Dynasty - approximately two thousand years ago. The compass was refined during the Song Dynasty and eventually spread to Europe by the fourteenth century. The invention of the compass allowed for much more reliable navigation and greatly aided European exploration from the fifteenth century onwards.

There were no major technological or economic breakthroughs between the Neolithic and Classical eras. Agriculture was a major breakthrough that marked the beginning of the Neolithic era, but we wouldn't see any other transformations of that significance until the Industrial Revolution of the 18th and 19th centuries. Civilizations and populations continued to grow as humans developed in terms of agriculture and were able to support more people. Furthermore, communication networks experienced major growth during the Classical era with development of networks like the Silk Roads, Indian Ocean network, trans-Saharan network, and Mediterranean network.

Aristotle was the head of Royal Academy of Macedon at the time of Alexander's youth, and thus became his tutor when he reached schooling age. Though this relationship has probably been overstated in modern culture, it's likely Aristotle did have some impact on the young prince. Aristotle was also instrumental for work in physics, biology, and many other fields. 

The first domed building to be built in Europe since the collapse of the Roman Empire was the Florence Cathedral, built by Filippo Brunelleschi during the Italian Renaissance. The Italian Renaissance was a period of rediscovery for European intellectuals, including ancient architectural techniques. The Age of Sail, in which millions of Europeans migrated to the Americas, Cortez's invasion of Mesoamerica, the unification of Germany, and William the Conquerors invasion of Britain had dramatic effects on Europeans and their knowledge of the world, but did not lead to the ability to build domed structures.

Gunpowder was invented in China.

Within a few hundred years it had spread all the way through Asia, including India and the Middle East.

When European nations invaded, conquered, and colonized the Americas, gunpowder was a staple of their arsenal. 

Hand cannons are the earliest types of handheld firearms and tend to lack both triggers and stocks; a burning torch would be applied, often by a different person than the one holding the cannon, to a hole bored in the cannon and lined with powder leading down into the barrel of the weapon where packed powder was ignited, pushing forward a projectile with enormous speed. Although over centuries, and from workshops as diverse as Medieval Switzerland, Sinkoku Japan, and Lincoln's United States, arquebuses and muskets were constructed with different attributes, in general all arquebuses and muskets have trigger mechanisms.

Wet gunpowder won't light, regardless the weapon system.

Hand cannons could be held by a single person.

Hand cannon technology was used all over the world, including but not limited to the Middle East and Europe.

Percussion cap technology occurred at the end of muskets being used by professional armies, long after hand cannons and arquebuses became obsolete.