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STL is supported as one of the mesh shapes in the <geometry> tag. An SDF file is still required.

The mass must be given in an <inertial> tag on a <link>.

There's an example of a model using an stl file in the cube_20k model here. Note that model is usind sdf 1.4. New models should be created with the latest version (currently 1.6).

There is a guided tutorial for creating Velodyne model with a mesh here. It says to create a collada mesh .dae, but the process is exactly the same for an .stl mesh. I recommend going through it.

It sounds like the origin of the .stl mesh is in the corner. I imagine you don't want to redraw your model at the origin in solidworks, so you can use the pose on the <collision> or <visual> to the right place on the <link>.