"I understand that pulses are generated at the transmitter's end to get to the receiver's end."
That is one of the ways we could transport information, there are other ways (which are usually more complex) as well.
"is it possible for the electrons to reach the other end of the wire if power is cut early from the source (a flow is already going through the wire)?"
I think in your view you see one (or a bunch of) electron(s) "carrying" the data to the other end. So to send something, an electron is pushed in by the transmitter and after a while it appears at the receiver.
That's not how it works! The conductor (wire) doing the actual transport is full with electrons. If I push in an electron then almost immediately (this change / wave travels at almost the speed of light) at the other end an electron will be pushed out. So the electron going in and the electron coming out are not the same one. And it does not have to be as they're all the same! So you would not be able to tell the difference anyway.
So if the wave is already travelling in the conductor and the connection to the transmitter is lost, the wave would still reach the receiver, no information is lost.
"is it taken into account with this kind of propagation delay when establishing data transmission connections generally?"
Yes, that is the electron(s) causing a wave in the conductor. This wave travels at about the speed of light. It is indeed an effect that needs to be considered for fast (high datarate) connections.
Example: take a look at a modern PC motherboard, more specifically the connections between the CPU and the RAM (memory). Note how some wires "wiggle" which seems pointless at that only makes them longer. But that's exactly the point, all wires need to be of the same length as we cannot have some bits arrive early and some later. All have to arrive at the same time so the traces need to be of equal length so all have the same propagation delay. There's an example in this question.