Let's say I have a normal topaz and I drop it into an item. I then find two more normal topazes (topazei?) and I want to combine them. Can I then remove the gem from that item, and if so how will it effect the gem and the item in question?
Yes, this is one of the Jeweler's abilities. It does cost gold, however, with more gold the higher the level of gem involved.
If you're not interested in getting the item back, you can salvage an item for the same result.
You can use the jeweler to remove gems from sockets.