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.
Just a short answer to get you thinking ...
When we connect two conductors together in a junction, do electrons actually move from one conductor to another?
Electrical current is the flow of charge. This is stated neatly in the equation $$ I = \frac {dQ}{dt} $$
where I is current, Q is charge and t is time.
Now consider a DC circuit with a switch. When you close the switch the current will be the same on both sides of the switch and in the same direction. It stands to reason then that the flow of charge must cross the contacts. If the resistance of the contacts is high - as is often the case due to the small contact area - then the contacts get hot as we would predict from the power equation $$ P = I^2 R $$ where R is the contact resistance.
Does it also affects the masses of the two conductors?
No. No additional mobile charge carriers are introduced. We just push the existing charge carriers along. A (poor) analogy is a bicycle chain: it transmits power but the number of links doesn't change and it's weight doesn't change. To compliment @jwh20's point about the speed of the electric wave relative to the speed of the charges, the effect of standing on the pedals is felt immediately at the back wheel even though it takes a second or two for the links to travel from the rear sprocket to the front chain ring.
Electrons have an appoximate mass of \$10^{-30}\text{ kg}\$.
Even so, when current flows in a loop, the electrons bump in one end and bump another out the other end of the conductor, so no change in mass occurs.
Electrons can jump across contacts when the voltage is different when contact is made.
When the current is high enough you can hear it and maybe see it. Such as plugging in a mobile charger. Since it is a current loop , the jump occurs on the last of 2 contacts made which is normally V+ by design.
This sound of the arc current is actually occurring so fast that it breaks the sound barrier and makes a tiny tick or a Big Bang such as thunder from lightning. This is simply from capacitance charge or discharge and the negative resistance from ionization of the arc between the conductors.