How does religion work?  Does it make cities change allegiance? Does faith determine its spread, and if so does faith do anything else?

Religion is created when your empire reach 10 population

Faith only use is to spread your religion, similarly to how influence spread

Religion gives you bonus from tenets (more are adopted if your religion has enough followers in ghe world) and converting others to your religion will increase number of follower, which will help to add more tenets. If they adopt your religion s state religion it will help diplomacy, if they dont you can get grievances to demand their territory following your religion.

You can also get a greivance to make them adopt your religion if you have the Heresy civic

In my current game, the religion I started - Babylonian Polytheism, BP - was quickly spread to my neighbours the Harrapans. At first I had a grievance against them as they owned a territory in which BP was the main religion whilst not their state religion (I assume), but I renounced this to keep relations cool. But then weird stuff started to happen. It seems the Harrapans adopted BP as their state religion, and have since picked the next two tenets for the religion and have built their own holy sites, reducing the number i can build! This is despite their being more followers in my empire - the empire that produced BP! Is this intentional??

I like the importance of trade routes too - thanks for the info. Any thoughts on the AI being able to "take-over" your religion?