I really don't understand what helps and what hinders the expansion of the cultural borders of my own and the AI's cities. Are there ways to make a city expand sooner, or do you just have to wait till the Civ4 program decides the time has come?
Also, when you have a city near and AI border, what can you do to keep the AI from expanding its borders and taking away tiles you need?
Say I build a new city near the cultural border of an AI city. Next thing I know, the AI city has expanded its borders and taken away some of the choice tiles, maybe even a resource, that I expected to have in my city. What can I do to counter this? Does it have to do with religion...such as having a religion in my city, priests or building temples or monasteries? Do those help? Or does it have to do with culture points, such as having a theater or a Great Wonder such as The Pyramids?

In my current game, I captured an 8 pop city from Hatshie, then a 10 pop city next to it. After a dozen turns or so, her next nearest city expanded and surrounded the 10 pop city I had just captured on all four sides! Is there anything I could have done to prevent this? Have a missionary ready before the capture and then send him in? Have some cultural Great Wonders I had built earlier? Build a quick temple in the city?

I should add that I set the custom rules to "No Culture Flipping of Cities" because of my ignorance of how culture works, which may be relevant to my question.

I will appreciate any help, or if you could direct me to an article somewhere in these boards...thanks.

To expand your cultural borders, your cities need to produce culture. The more the better.
Obelisks, temples, libraries, theatres, wonders, specialists, etc. -- there are tons of things to help you produce culture. Religions help build culture.

As far as expanding your borders are concerned, you'll need to produce enough culture in any given city. Go to the city display screen, and look on the left-hand side. Just above your build queue is a purple bar that shows the cultural progression of that city. Hover over it with the mouse, and it will show you how much culture the city has to produce to the next border expansion (eg. if it shows 60/100, then the city has to build 40 more culture for the borders to expand). Every time your borders expand, the bar resets, and you start working toward the next level/expansion.

The best way to keep an AI for taking away tiles is to out-culture them. If two cities are after the same tile, the city with the stronger culture level will eventually win that tile. Or, you can just capture/raze the offending city.

One thing you can do with newer cities being pushed by the AI, or captured cities, is drop a culture bomb with a Great Artist. Not guaranteed to work, but it can help.

I hope that helps. If you have more questions no doubt I or someone else can try to help you out.

(Edited for spelling)

Thanks for your very helpful suggestions.

Just happy to help.