Can you explain how influence works with reputation and the bathrooms? How are they built and how much reputations grants influence? How do I get reputation for our town?
That depends on the settings of the specific server. The general way it works is that for every X culture you gain 50% towards the maximum influence, e.g. a diminishing return, as you always get another 50% of the remaining span of influence you can still achieve.
Bathrooms give direct culture points instead of housing points when used on cultural deeds, though that is limited.
The main way is to make museums with pictures and get them rated by other players. Reputation from players in the own settlement does not contribute much to culture, you'll need people from outside.