Can Sid Meier's Civilization (2010) be played cooperatively?
Civilization can certainly be made a 2x2 team game.  The rules you wrote will work.  One slight addition:


Rules and culture cards referring to "an opponent" (such as A Gift From Afar or Disoriented) refer only to players on the other team.  If it refers to "another player", it can be used on a player on the opposing team or your ally (such as Exchange of Ideas).


Some things that would make it more strategic or interesting:


A players gains the benefit of his/her allies technology (if your ally has a technology and you don't, you play as if you had that technology, so can buy buildings/units on it, use it's abilities, and gather coins as if you had it).


Whether or not you allow trade between non allied players is up to you; I think it is more interesting to leave that as allowed and just rely on the competition between the teams to discourage it.

Whether allies start on adjacent corners or opposite corners is also of interest, though I think adjacent corners will be a better game play experience.

Playing it as a team game might change the balance of some of the win conditions.  Military victory probably gains some strength over the economic victory, as military cooperation is easier.  That said, having an ally that can just feed resources or culture points to the other could actually make the cultural victory distinctly easier.  And technology sharing probably makes the economic victory easier.  That there isn't anything obviously unbalanced about this means that it is an interesting variant and would require play to determine.





The original question could have been read as to whether or not Civilization the Board Game could have been played as a full co-op board game.

One of the fundamental tenants of a fully cooperative board game (like Pandemic) is that there is a way for all players to lose.  Civilization the Board Game has no built-in mechanic for all players to lose, so would not work as a full co-op game without serious modification.  If you are looking for something that feels vaguely similar but with more aspects of full co-op games, try Archipelago.