-> Lennier, Pangaea
So, it's about culture. More than that it's about the culture status of adjacent individual cities, namely Rome and Seville.
I can only get more culture by building cultural buildings, and, I've built the "ordinary" ones (theatre, palace, castle), I'll have to the "expensive" ones, e.g., Broadway, Hollywood.
Apart from the Theatre, those aren't exactly ordinary culture buildings. The Palace is only in your capital, and castles suck.
So late in the game, getting up the culture producing buildings wouldn't help a great deal, but early on, every little bit helps. Library, Monasteries, Temples, Universities, your state religion, and after 1000 years, the buildings will double their culture output. Early libraries will then give you 4
pretty early in the game.
That said, culture isn't really something you should focus a great deal on, unless you want to win by culture. If you're in a dire situation where you could lose food, it could make sense, but in more normal circumstances it doesn't matter all that much if you lose a mine or a farm, and it's not going to happen very often anyway. Your culture tends to be pretty strong in the BFC of a city, and unless an AI city is so close that it too can get those same tiles in its BFC, you're probably not going to lose them.
Of course there are more special circumstances, like an AI pumping out wonders in a border city, having the holy city there, getting Sistine, and things like that. But in 'usual' circumstances, you shouldn't be mass producing culture in your cities for fear of losing tiles. You can always ensure you gain those tiles by taking the border cities anyway
About the longbows. In one of the pictures there is a longbow outside of a Roman city. I assume he has better defenders than that so late in the game, but if he doesn't, you could so easily take his stuff.
Also, Cannons are fantastic. Try to go to war with them. They'll smash up just about anything in front of them, even Rifles when well promoted, and then you can take cities with almost any clean up unit you have access to. If you for instance get Cannons attacking cities defended by Longbows, even Maces, you will often see 80-90% chances with the first attack. They're simply so strong that you can wage war with them and barely lose a single one. Artillery is even better of course, but comes much later.
Three units that basically mean a paradigm shift in warfare are Cannons, Rifles and Cuirs. They're massive upgrades on what came before them.