"The dog walks from there."
Translation:Der Hund kommt von dort.
"hier", "da" and "dort" are used to indicate near, middle, and far distance, but they're used pretty flexibly. So far as I can tell, if they're used in the same sentence, it will mean "here, not quite as close, and further away" -- which is all going to depend on context.
"da" is also sometimes used to mean "right here", but I think the context is usually discovery -- "your keys are right here!" "da sind deine Schlüssel!".
Googling the phrase "läuft von da", I see it used a good bit, but generally in descriptions of point-to-point courses or situations: "Lasst euch... zur Frühlingsinsel „teleportieren“ und lauft von da aus etwas südlich und dann nach Westen." -- "let you teleport to Spring Island, and go from there somwehat southerly and then to the West"
Note that the subject has arrived at Spring Island, and is going on from the location it is at -- I think that is why "da" is used, not "dort".
So I suspect in the exercise, "kommt von dort" is what you would use -- "laufen" and "kommt von" both imply distance, and there's nothing else to provide any kind of closeness.
(Correction from actual German speakers welcome...)