I see that sometimes my territories get converted to another empire's culture.  Other times my culture takes over another empire's territory.  What is the point? Does this have any impact at all on gameplay? Is there any effect as a result of this?

A cause for war support. If you control other empires territory you gain any influence that territory made as far as I can tell from the culture lens. Subsequently you lose any influence you generate from that territory if you are taken over.

It makes it extremely cheap and easy to conquer cities in your sphere of influence, but if no war is ever declared it has no impact.

Also if a certain part of your territory is dominated by a different culture you will receive Culture demands to make your Society closer to that dominant culture. If you decline you will receive a -50 Stability hit for 10 turns.

Faust wrote:

 but if no war is ever declared it has no impact.

War is not all that cultural conversion has an effect on: when you you own a territory of foreign culture, osmosis events  (both cultural and scientific) can happen.

Everyone who is complaining about war support needs to know this.

You can conquer so many cities if you follow these steps:

1) Culturally convert them by having high influence

2) Get the grievance for oppressing my people

3) Demand it (it moves from grievances to demands).

4) Go to war... when you win you get that city, regardless of war score. You can do this for as many cities as you want before the war starts.

If you're strong enough militarily and have cultural dominance, you can demand territories from weak empires and they may give it to you outright, without a war. I've done this a number of times.