The importance of science is annoying.
It's not as if the Maya came up with the zero because a lot of smart guys sat down in a circle and said "let's invent something new." Romans didn't create their grand structures as research projects. Culture guides science, but there is no importance of our civ's culture on science (except the rationalism tree, but, trust me, the Romans were mixing concrete WAY before the Renaissance.)
Religion has quite a bit to do with science. People look for answers and discover things. Heck, Europe was passively destroying knowledge from ancient Greece and Rome in order to write religious texts, while the Muslim work was actively saving many of the same texts for religious (educational) purposes. In the game, there are a total of two beliefs that can give science.
War has had a massive impact on science. Again, people didn't sit around and say "hey, let's make iron stronger just because iron is cool" or "you know what, I want to create a bomb from that radioactive stuff because...fireworks!" only to later see the use of that technology in weapons. In the real world, war leads to an insane percentage of research. In the game, science is the primary, war comes as a result.
Science doesn't happen for the sake of science, ever. It happens because pure science is valued culturally (as in ancient Greece), religiously (as in the middle ages), or for practical reasons (such as ancient Rome). In the game, science happens because...science is science and it's how you win.
Yeah... You are mixing here some facts, common beliefs (aka myths) and misconceptions. All this comes from the fact that you are looking at things from the perspective of modern man and you confuse general science, the process of discovery and advancement, with science defined in modern terms, i.e.: organized application of scientific method undertaken in order to achieve previously defined goals.
First of all, war does not cause technological improvement - it puts bigger pressure on people to get technological advantage over others. Using the approach from your post: "I guess we could use something stronger than copper and it would make several aspects of our lives slightly easier, but it's too much of a hassle now... The good, ol' copper will do. But wait! The evil Ruskies are coming to eat us? In this case, let's better invent this UberCopper!"
Also, culture, religion and science are not three different aspects of civilizations. Such separation is a trait of the modern Western civilization. It doesn't exist in Arabic word and is only nascent in China or in India. Most of our (Western) problems with Muslims are caused by the fact that our societies can't comprehend this fact. It's silly, because this separation was born in Renaissance and became a wide-spread fact only during French Revolution. The XXth century was the first century when such view of the world became natural to us.
I don't really have time now to address all the misconceptions in your post. Even the paragraph above requires a much broader perspective... Just one more thing:
"Science", as we call it today, did happen for science's sake, or in other words: it was valued culturally, as you put it, even though you make an incomprehensible distinction between these 2 things. Value, or the whole axiology in general, is a derivative of culture, so if we attribute value to anything, it's always a cultural process.
Ancient philosophy often shunned practical applications of "science" and even nowadays we very often work just to expand our understanding of reality. Large Hadron Collider is a great example of this fact. We have no religious or immediate practical reasons for that; we are trying to validate a theory.
PS.
Technological advancement in medieval era had theological background? That's something new to me and I have a decent background in medieval philosophy. There's a reason why we call it "Christian dark ages". Maybe you were referring to Muslim world or the Far East?