Life in Rome? In the western provinces e.g. Britannia, Germania, Gaul, life is by far better than it was during the medieval times or even in Antiquity. Due to the intellectual and architectural wonders of Rome, the regional capitals for the provinces have rapidly improved with their new technology. Trier is a thriving Roman city with elements of its former tribal past, with its Military school being one of the greatest in the world, let alone the Roman Empire. The culture of these regions is heavily Romanised now, people see themselves as Romans rather than Britons or Germans by now. Life spans have drastically increased, on average a person would live to 42, compared to the 30's and 29's of the Medieval Era and Old Pax Romana respectively. Almost every city in Italy has been developed massively, Napoli, Capua, Mediolanium and Genua are all large cities of between 400-800 thousand, with Rome herself having around 2 million. In the east, Greek culture is still strong. although its slow merge with Roman is hard to ignore, texts in Greek have been translated to Latin, allowing even the average Briton or German to listen to the words of Aristotle.
Islam, while not as strong as in our timeline, still dominates arabia, central asia and Persia. Islam is very similar apart from that, still has a sense of Jihad if a muslim nation is invaded, however due to there only being two truly powerful muslim nations, and those two nations having an alliance it isn't at all surprising.
Jews are a large minority group in Judea, after the Great Jewish revolt the Jewish people and their respective leaders decided they couldn't win a war with Rome, and instead decided to spread jewish culture and tradition as much as they could, to try and make the region more God's province on earth. Jew's are a relatively respected minority, especially in our timeline.
The Nordic Union, while that name needs revising is a more centralised version of Slavia, with a 'Great King' as their ruler. After the defeat of the Danes most tribes saw the danger, with the Sveas taking centre stage to the unification. It took a while, and due to Rome being distracted with Slavia the unification wasn't slowed down by Rome, so in the end its only been unified about a 100 years before the start of the colonial period, with the norse discovering America. They are norse, with some slavic, roman and christian minorities.
They only just discovered them, however with the industrialised ships and better weaponry Rome will probably colonise very quickly, especially considering how in our timeline it was fast with only England France and Spain (sorry Netherlands) taking north america. There isn't a specific plan, they are still coming up with ideas, however in the end it might just be a rush.
Romans definitely trade, however they do not like to trade ideas. Envoys would rather die than give up Roman steamwork and guns. Norse have good trade relations with slavia, and the fact they've noticed the Second Sassanids at all is impressive, although trade relations are just starting. China dominates east asia even more than our timeline. If you see a ship in East Asian waters trading you'd probably be right to say its Chinese. The Indian principalities trade with almost anyone, whether Roman, Sassanid or Chinese, however each one has its own preferable partners. Finally, the Gupta trade lots with the Romans, hoping to ally them incase of a Sassanid Conflict.
Holy jesus that was alot of writing.