If the universe keeps expanding will it hit itself eventually?

That's a really good question that no one quite knows the answer to. You're right in saying that there have been theories that space has got a curvature, but the most recent results show that it hasn't got a curvature at all. It might be completely flat, or it's so close to flat that we can't tell the difference between curved and flat. How do they work this out? It's pretty tough but it's all to do with the speed at which that things are flying away from us. If you can imagine measuring the angles of a triangle on a flat piece of paper, they add up to 180 degrees. On a curved space that's not the case. So we use these geometries to work out what exactly this curvature is. As I said, at the moment it looks like everything's completely flat, or at least very very close to it.