The lead field matrix is another name for the gain matrix produced by the forward model.
It is a computationally optimized version of the MNE equation you are quoting in #3. @jcmosher can tell you more about how it is derived more specifically.
The kernel is not an image, it is a matrix operator applied to the sensor data to produce an estimate of source time series at every cortical location. It is the large matrix multiplication to the left of M (the sensor data) in the equation in your point #3.
Hope this clarifies.