Do electrons actually jump across contacts?
Yes, actual electrons do move through conductors and yes, they cross boundaries like connections between two wires.
While electrons do have mass, there is no net addition of electrons through a conductor.  If N electrons are added at one end you will get N out the other end for a net mass change of 0.
Keep in mind that electrons move in a reverse direction (i.e. from - to +) which is the opposite of the conventions we normally use for current from which is from + to -.  Also note that electrons move quite slowly through conductors even though the electric current moves much faster.  The electrons themselves move at something like 1 mm/sec in a copper wire.