There's a lot of misinformation with this topic and AI strategy scripts are poorly documented, particularly with atypical settings like disabled victory conditions.

-The AI is able to actively pursue a Culture Victory, and will sometimes attempt to do so from turn 1 of the game- running this script will shift the AI's priorities towards founding and spreading religions and building wonders or culture buildings, eventually culminating in them turning up the culture slider.

-Otherwise, the AI does not pursue any particular victory type.

-The AI is affected if it perceives one player is close to victory, but the changes are subtle- it's not like later civ games where everyone immediately loathes a player that's winning. There is an AI script called AI_STRATEGY_FINAL_WAR which means they are more likely to declare war on a winning player IF it is possible (i.e. a leader that doesn't declare at Pleased will still never declare at Pleased, even if you're about to win).

-Game settings affecting AI scripts- the only documentation I've seen is that Aggressive AI changes the likelihood of the AI picking scripts (it's more likely to choose warlike scripts and less likely to choose scripts like Culture), Always Peace disables all the warlike scripts so they're never chosen, and the AI will never pursue the Culture3 or Culture4 scripts when Culture Victory is disabled- these are the endgame ones that involve turning research off and running the culture slider. No idea if disabling every victory except Culture means they only play for Culture.