How do I add a Matrix room to a space? Can I move a room from one space to another?

A space is just a special room that serves as a container for other rooms, so a room can belong to any number of spaces. If you are an administrator or owner of the space, you can change the rooms listed in the space whenever you like. In the web or desktop versions of Element, in the "Space Home" accessed by clicking on the space name, you should see "Add" and "Remove" buttons to manage the rooms listed in the space.

If you don't have administrator privileges in the space, you will need to contact someone who does and ask them if they're willing to make the change.

If you want to move a room from one space to another, you can simply remove the room from the current space and add it to the new space.

If you change the spaces associated with a room with space-only join restrictions (privacy set to "Space Members" in Element), you should also review the room settings afterward to make sure users belonging to the new space(s) are allowed to join the room.