First, get your terminology right. The schematic has wires, the PCB has tracks (often called “traces” in other PCB design programs, I don’t know why KiCad settled on “tracks”, although I find the name more logical myself) Using the right terminology reduces misunderstandings and reduces post count for attempting to get more clarification.
You have turned PCB Editor / View / Drawing Mode / Sketch Tracks on. If you turn this mode off again, the tracks will be filled again. It’s just a visual thing, nothing more.
Why do you think so? I see a ratsnest line in the lower left corner:
And this indicates that the two track segments you drew are connected to pad 8 of that other footprint. If you draw another track segment onto that pad 6, then that ratsnest line will disappear, but others for still unconnected pads will still remain. I suggest you draw some more tracks before you make any conclusions about this.
Indeed. When you move a Footprint (please spend some time on terminology) it just moves the footprint. You can drag a footprint (with the d hotkey) and then KiCad attempts to keep tracks connected, but often it can not drag all the tracks without creating DRC violations. In such a case you can finish the drag witn [Ctrl + Click] to accept the final position with the DRC violations, and then clean up the DRC violations.