Is this what is causing the "plague" that the world is now experiencing?

irrelevant, there are no creatures, "plague" doesn't matter, it's used as a plot device to make people not trust each other, to have a reason why they are in a forest and why they don't trust people and would be willing to kill a family.

and drag it back to the house all while everyone else was sleeping.

he did that but reasons for why dog is dead are irrelevant, he killed it sleepwalking, dog fell in a ditch and cut his sides open, other guy's gang members killed the dog... dog plot exists as a device for two families to split apart.

Also, how did their son get sick?

when he got the dog, when he touched the kid, when he was in grandfathers room.. doesn't matter, that's the thing, you don't get to know who was responsible, was it the other family? was it himself?

Movie is about two families not being able to trust each other, it's not about anything else, you should ask questions about whether what happened in the end was warranted, whole movie was about what would you do at this point with information you know.