It sounds like you may be trying to use a transparent background layer. It's not that the tile is repeating wherever you use the mouse, but rather there's no background to replace the temporary image of the tile that was being displayed where the mouse was. You should create a layer with a non-transparent background. As soon as some layer is covering the map with some tile, it will start to work more like you expect. You can set a background tile when you create the layer.
And yes, you can have multiple layers. Just keep adding layers to the same map.